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The probability that part of a set of equicorrelated normal variables are positive

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The probability that part of a set of equicorrelated normal variables are positive
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Hoffman, Thomas Ray, 1945-
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Thesis -- University of Florida.
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Bibliography: leaves 98-99.
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THlE PROBAOBILITY THAT PART OF A SET OF EQUICORRELATED
NORMAL PARABLES ARE POSITIVE














BY

THOM11AS RAY HOFFMANX






















ADISSERTATIONK PRESENTED TO THE GRADUATE COUNCIL OF
Tji-F U~dVERSITY OF FLORIDA IN PAR7i'IAL
7F7LF7LLM11ENT OJF THE REQTUIprE ,Nrs FOR THE DEGI2FE OF
DOCTOR OF PHILOSOPHY




U NlVE R ST 71 OF FLORIDA
1.972








































TO MY PARENTS





















ACKNOWLEDGMENTS




1 would like to express my appreciation to my major professor, Dr. John Saw, who suggested the topic of this dissertation, and who was always available for assistance. Appreciation is expressed also to the other members of my supervisory committee, Professors R. L. Scheaffer, P. V. Rao, and Z. R. Pop Stojanovic.

Also, I would like to extend my thanks to the other members, faculty, students, and staff of the Department of Statistics. They made my stay at the University of Florida both rewarding and enjoyable.

The manuscript was typed by Mrs. Edna Larrick. Her patience and assistance through those final pre-deadline weeks will always be remembered and appreciated. Also, I would like to thank Mrs. Deborah Ingram for her excellent work drawing graphs.

Finally, I express deep appreciation to Professor Paul Benson~ of Bucknell University. Without his guidance and encouragement I would never have entered the field of statistics.

















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS...................... . ... .. .. .. ....

LIST OF TABLES...........................vi

LIST OF FIGURES...........................vii

ABSTRACT...............................viii

CHAPTER
1 INTRODUCTION.........................1

1.1 Introduction.......................1
1.2 Definition of P m(P) ..................2

1.3 A Transformation Simplifying P rm(P)...........6

1.4 Summary of the Results of This Dissertation . 12

2 AN EXPRESSION FOR P rm(p) INVOLVING TCHEBYCHEFFHERMITE AND LEGENDRE POLYNOMIALS ..............14

2.1 Definitions and Properties..............14
2.2 The Fundamental Result..................18
2.3 The Integral J k(P) ................ 2

3 NUMERICAL RESULTS ......................26

3.1 Exact Results.....................26
3.2 Evaluating the Integral.....)...............31

3.3 Computing P m(p)....................38
3.4 Accuracy of the Results.................42

4 APPLICATION: A TEST FOR NORMALITY .............44

4.1 Introduction ......................44
4.2 The Null Distribution.................47
4.3 Approximations to the Null Distributior. .......49







iv












TABLE OF CONTENTS (Continued)

CHAPTER Page
5 OTHER METHODS OF EXPRESSING P (p) . . . . . 62

5.1 Introduction . . . . . . . . . 62
5.2 A Power Series in Rho . . . . . . . 62
5.3 A Series Resulting from an Inverse
Taylor Series Expansion of g(u) . . . . 73
5.4 Using Moments of Extreme Order Statistics . . 82

APPENDIXES . . . . . . . . . . . . . 84

BIBLIOGRAPHY . . . . . . . . . . . . 98

BIOGRAPHICAL SKETCH . . . . . . . . . . . 100










































v
















LIST OF TABLES

Table Page

1 Standard Deviation of Y....................48

2 fhe Cumulative Distribution of Y, n =19...........51

3 C (P ,2)..........................57

4 C (p )3).. ...................... .......58




6 Moments of Y V- Y, n = 9....................61

7 Error Involved in Computing P m(p) when the Series
in Rho Is Truncated after Five Terms............72

8 Values of a ., j =0,2,... ,22 .................78

9 Values of vk (in), (1/3), $ (1/4), k k k
k = 0,2,... ,22, m = 10...................80

10 The First n Terms in the Series P (P),

m = 10, p = 1/3, 1/4, n = 0, 2,... ,22............81


























vi
















LIST OF FIGURES

Figure Page


2 h-1
1 The function h Lk[2G(u) -1] Le Tu p = 1/10 . . 32


1 r 2 h-1
2 The function hi Lk [2G(u) 1] e 2, p = 1/2 33


2 h-1
3 The function h Lk[2G(u) 11]e p 9/10 . 34u



u 21 -h-1
4 The function (-h) i Ekf(u)] [e u2 -hp=- 1/2, p=-1/10 .................. 37































vii












Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy


THlE PROBABILITY THAT PART OF A SET OF EQUICORRELATED NORMAL VARIABLES ARE POSITIVE

By

T'homas Ray Hoffman
March, 1972



(Thairma,: Dr. J. G. Saw
Major Department: Statistics


The probability that part of a set of equicorrelated normal variables are positive is_ defined by a multiple integral expression involving the multivariate normal density function. Although much research related to this integral expression has been published, most results do not include a practical method of its evaluation. Also, when the correlation is negative, no direct method of evaluating the integral expression is available. in this paper we discuss several methods of expressing the integral. One of these expressions, valid for both positive and negative correlation, is used to obtain numnerical results.

A transformation is used to simplify the integral expression for the probability that part of a set of equicarrelated normal variable are positive. Then the probability can be written as an integral involving the real normal distribution function when the correlation is positive, and the complex normal distribution function when the correlation is negative. For positive correlation, this integral expression has been used by other authors to obtain numerical results.


Viii












Next, we use a result connecting the terms of a binomial series with Tchebycheff-Hermite and Legendre polynomials to obtain a finite series expression for the probability. Although the general term in the series involves an integral which cannot be evaluated in closed form, this integral depends only on the correlation and can be evaluated by numerical integration for both positive and negative correlation. The numerical results are Included in the appendixes.

As an application, the number of observations larger than the sample mean is used as a test statistic for testing the hypothesis that a population is normally distributed. The small sample null distribution is derived and numerical results are given. Approximations to the null distribution are also discussed.

Finally, we discuss three other methods of expressing the

probability that the entire set of equicorrelated normal variables is positive. Two of these methods express the probability as an infinite series. However, in both cases the convergence is quite slow. The third expression, involving moments of extreme order statistics, can be used for obtaining numerical results only for limited positive values of the correlation.


















ix















CHAPTER I


INTRODUCTION




1.1 Introduction


in a recent paper, David (1962) suggested using the number of observations larger than the sample mean as a test for the homogeneity of a random sample. Assuming a normal population, he showed that the proportion of observations larger than the sample mean has an asymptotic normal distribution. However, David did not discuss the small sample distribution for the test statistic.

The work on this dissertation began in search for the small sample distribution for David's statistic. However, this work soon led to the more general problem of finding the probability that part of a set. of equicerrelated normal variables are positive and, in particular, the problent of evaluating a multivariate normal integral expression for the probability that all the variables are positive.

Much research related to the multivariate normal integral has been published. Gupta (1963b), in addition to an excellent survey paper, gives a complete bibliography of articles related to the multivariate normal integral. However, only a few of these articles offer a practical met-hod of evaluating the integral. Also, although Steck (1962) gives a relation connecting the results for positive and negative correlation, no direct method of evaluating the integral has



1







2




been obtained when the correlation is negative. Only two authors, Ruben (1954) and Gupta (19630, give numerical results.

It is the purpose of this dissertation to find at least one method of evaluating the multivariate normal integral that works for both positive and negative correlation and that can be used easily to obtain numerical results. Then the small sample distribution of David's statistic can be given as an application to the more general problem.


1.2 Definition of P m(p)
r:m


Suppose the m variates Xl,X2, ... ,X,, each with zero mean and unit variance, have a multivariate normal distribution. Of interest is the probability that exactly r of these m variables are positive. If the variables are mutually independent, the problem has the binomial solution (Mr) G2). However, when the variables are dependent, no simple solution exists. In this paper we shall consider the case when the variables have common correlation p, --1 < p < 1. P (P) M-1 r: m

will denote the probability that exactly r of the m variables are positive. That is,


P rm(p) = E P(Xi1 > 0 .... Xir > 0; Xir+1 < 0,... ,Xim < 0). where the summation is over all partitions [i .... ; I ,... ri m of the set [1,2,... ,in. Since X ,X2P Xm are identically distributed, the above equation may be written as

pr(P) =(m P(X > ,. A,X > 0; Xr+1 <0'''A <0)



Letting g(x2,2 xm represent the density on XI,2''.. Xi

we have






3




(1.2.1) Pr (P) = ' g(xl ,x2,...x )dx1,dx2,... dx
x'> o x. 1 1
i< r i>r = g(XlX2,...,x ) dx1,dx2,...,dxm'

[r,m]


where [r,m] will be used to denote the range of integration


(x. > 0: 1 < i < r; x < 0: r+1 < i < m)
1 1

In the case r = m, we will, for convenience, write

P (p) rather than P (p).
m m:m

A first approach to the problem might be to write down the density function g(x1,x2,... ,x ). From multivariate theory we have m 1 1 x -1x
2Vx
2 I iV 2 2 (1.2.2) g(x1,x2,... xm) = (2rt) 19 e
C- < X. < C i = 1,2,...,m, where x denotes the vector (xi,x2,... X ) and V is the dispersion matrix given by

1 ... p
p 1 ..p Vi


p p ... 1


Due to the simplicity of the dispersion matrix, the determinant IV!
S-1
is easily evaluated and the quadratic form x V1 x has a simple scalar representation. In fact, it will be shown that


(1.2.3) IV, = (1 -p) m-[l+(m-l)p],







4




1-+ (m-2)p -P "

1 1 l(m-2)p ... -P
(1.2.4) V (l-p) [+ (m-1)p]


P ... I+ (m-2)pj


Since the determinant of a matrix equals the product of its

latent roots, (1.2.3) can be proven by finding the m latent roots of V. If X is a latent root of V it must satisfy (V XI)y = 0


where 0 represents the (mxl) vector of zeros, for at least one non-zero vector y. Letting % = (l-p) we note that the matrix (V %I) contains only one distinct element. Therefore the rank of (V %I) is one and there exist (m-l) non-zero vectors y satisfying (V XI)y = 0. Hence, (1-p) is a (m-l)-fold latent root of V. To find the last latent root we note that the trace of a matrix equals the sum of its latent roots. Since the trace of V equals m, X = m (m-1)(1-p) = 1 + (m-1)p,


and (1.2.3) is proven.

To prove (1.2.4) denote V_ by A = (a0). Then A must satisfy

m
Sv~j aj = 1,
j=l

m
Z va -0, a .
j=l ai aj












Substituting for v in the above equations, we have (1.2.5) p E a + a = 1,



p E a + (1-oa + p a = 0.



Subtracting these last two equations, we have


1
(1.2.6) a a


It is clear from the preceding equations that a is independent of a and 0. Hence, equation (1.2.5) simplifies to (1.2.7) (m-1)pati + a = 1.


Finally, the solutions of equations (1.2.6) and (1.2.7) are


a-P
a = (1-p)[l+(m-1)p] 1+ (m-2)o
B= (1-p)[1+(m-1)p] '


as was to be shown.

Using the scalar representations of IVI and x 'V-1x in the density function, (1.2.1) may be written as m m-1 1
(1.2.8) P (p) = ) f (2) 2 (l-P) 2 [1+(m-1)p]
r: m r
[r,m]

r m 2
M 2
[1+(m-2)p] x 2p Z x.x.
i=1 i exp L- dx ...dx.
2(1-p) [1+(m-l)pl 1 m


Unfortunately, although void of matrix notation, the above representation of Pr:m(p) is not too desirable for obtaining numerical results. A more workable form of Pr:m(p) is needed.







6




1.3 A Transformation Simplifying P r:m(P) r: r



Consider the m variables X ,X ,... ,X' defined by
X =1 e

1 1 o

X2 = Y By '2 2 o

X = Y eY m m o


where the (m+l) variables, Yo ,Y ,... ,Y m, have a multivariate normal

distribution with mean vector zero and dispersion matrix I, and e is

an arbitrary constant. It follows that X',..., have a multivari1 2 ~m ate normal distribution with


E(X) = E(Y.) @E(Y )
1 1 0
= O,
o2
Var (X) = Var (Y.) + 2 Var (Y ) -29 Coy (Y.,Y )
1 1 O 1 0
2
= I+ 82

Coy (X.,X.) = Coy (Y.-BY ,Y.-BY )
1 3 1 o 0
= Coy (Y.,Y.) e Coy (Yi.,Y ) 1 3 1 0
2
B Coy (Y ,Y.) + e Coy (Y ,Y ) 0 3 o 0
= 2


Therefore, assuming p > 0, if we define 9 by
*
e =


X X/ X
2 2 -1 1 2 m
B (1+2) equals p and the variables 1 2 ... m
(1+8 )2 (1+92 2 (1+92)

have the same distribution as X1,X2,... ,Xm defined in Section (1.2).


That is,







7





r
Var Var (X')



1 2 /


9
(ieO] coy (X. ,X.)
2 2 1+013



82
1+ 2




Hence, we have

p (~m (0 (r) P(X 1>0,.. 'X r>0; X r+I. <0 'Xm < 0)









r+1








=(.M) P(Y-GY >0,... e 0


yr+ -eyo

g(y) e 1_Yc




G(y) f g(t)dt,







8




respectively. Hence, writing P r:m(p) conditional on Y = y and inteSr:m o

grating over y, we have
Pr() = (M) NY (I> Yo"'..'Yr> GY ; Yr
r:m r 1 o r o r+1 o

...,Y

Finally, using the independence and identically distributed properties of Y1,Y2''.... Ym' we have
OD

(1.3.1) P (p) = () f [1-G(8y)] r[G(Gy)]m-r g(y)dy.
r:m r -C


Results similar to (1.3.1) have been given by Ruben (1954), Dunnett and Sobel (1955), Moran (1956), and Stuart (1958).

Although the expression for Pr:m(p) given in (1.3.1) was derived, assuming p >0, Steck and Owen (1962) have shown that it also holds for p <0 by defining G(8y) in the complex plane. For p <0, 8 is an imaginary number and can be written 9Cp where





(1-P)


Then G(ey) equals G(i rPy) and is defined by integrating along a path in the complex plane parallel to the x-axis from -m + iyy to + ipy. That is,

2 2
(1.3.2) G(py) Y 0 e g(t)dt.
(1.3.2) GCLcpy = e e e (t)dt.








9



The proof of (1.3.1) for p <0 consists of showing that the right-hand sides of equations (1.2.8) and (1.3.1) are identical. First we note that


1 GCLCy) =1 J' j1S e- '9 g(t)dt




= ek e--t'Y g(t)dt




+ e7 Yf e-"Wg(t)dt
0



=e y I eL4t'y g(t)dt.
0


Then using (1.3.2) and writing the right-hand side of equation (1.3.1), say R, as a multiple integral, we have


R (~; [1-GQCPy)]r[GCLcpy)]mr' g(y)dy r 22 r

t. t0


m-r 1 Lp


j r 1 rj~d



m
CO -2 2 -jcxpy Z t.

j eg(t .. g(t )dt dt]y~y

1* g My**dy



L[r, ml







10




-1
Since p >
m-1

2 -mp
1

< m
m-1+l

= 1

and the integral
m
= 2 -y t.
2,2y2 j=1
e e g(y)dy



converges. Therefore, interchanging the order of integration is permitted and we have

m

(m) ( P =1 3 1 (1-m2 y
R r~j [T~Y-i 1 t.)2

[r,m] ,/-2-T


g(t ) ..g(t )dt dt .
1m 1 m


The integral in brackets multiplied by (1-np2) is the characteristic function of a normal random variable with zero mean and variance (1-up2 -1 Hence,


m 2
(CP E t.)
m t2 j=1l
m t
2 2-y j=l 2(1-n )
R = (2n) 2(1-mq2 ) e e dtl...dtm
L r,m]












Substituting -p(l-p)- for cp Z 2

m]

P 2. j-1





e 2[1+(m-l)p I dti*dtm Finally, making the transformation X.

S (1-p )2

we have



R =()F(27T) 2 (1) 2 [1 l-)f

[r, m]




exp ([+m~)~ l j kP 2 Jjdx... dxn
2(1-p ) 1+ (m-l)p] and the proof is complete.

The results of this section for positive and negative rho can

be summarized in the following lemma. Lemma 1



rP (m) [1-G(Y),r [G(ey)]m- g(y)dy

where


e- P







12




and is written as ic when p <0 with 9 defined a.s




(1-p)i

The functions g(y) and G(Oy) are defined by


1 2
g(y) 1 _jY2
g(y) = -< e y




Sm g(t)dt, p > 0
G(By)

e 2e~2 e g(t)dt, p < 0




1.4 Summary of the Results of This Dissertation


In the next chapter Lemma 1 and a resuLt connecting the terms of a binomial series with Tchebycheff-Hermite anrd Legendre polynomials are used to obtain a finite series expression fior Pr:m(p) valid for all allowable values of p. For p <0 the results reduce to a workable form once the real and imaginary parts of G(iy) are isolated.

In Chapter 3 it is shown how the results of Chapter 2 can be programmed to obtain numerical results. Since Pm (p) is simpler than

Pr:m (p) for computing purposes, a result expressing Pr:m (p) as a sum involving P.(o), j = r,r+1,.... m, is proven. 7The accuracy of the computed results is verified by comparison with exact results for special values of m and p and with the results of Ruben and Gupta.












An application of the results of the first three chapters is

given in Chapter 4. The statistic suggested by David is used for testing the hypothesis that a population is normally distributed. The null distribution is discussed and numerical results are given for sample sizes not exceeding 22. Approximations to the null distribution are also given.

Finally in Chapter 5 we discuss three alternative methods for computing P m (p). However, none of these methods can be used to obtain numerical results as readily as the method discussed in Chapters 2 and 3.
















CHAPTER 2


AN EXPRESSION FOR Pr:m(p) INVOLVING TCHEBYCHEFF-HERMITE AND LEGENDRE POLYNOMIALS



2.1 Definitions and Properties


Let ck(r,m) denote the kth order Tchebycheff-Hermite polynomial orthogonal on r = 0,1,... ,m. Then ck(r,m) can be written (see, for example, Plackett, Sec. 6.5), as


(2.1.1) c (r,m) (k!) 3 k / r > (2k-ji\ m-k+j)
k (2k):. (-1) k-j} k j / '
j=0


and satisfies the following three properties:

m
(2.1.2) Z ck(r,m) = 0, k = 1,2,...,
r=O m
(2.1.3) E c.(r,m)ck(r,m) = 0, j k,
r=O


m (k) 2 (m+k+l)
2 \2k+1
(2.1.4) ck(rm) 2k
Y k/2k\
r=O kk )


Also, let Lk(t) represent the kth order Legendre polynomial in t. Lk( t) is given by (see, for example, Abramowitz and Stegun,
kk
k 2 "
Chap. 22) the coefficient of s in the expansion of (l-2ts+s ) and can be computed from the recurrence relation



14









L M11 L (t) = 1
0
(2.1.5) L(t) = t

2k-1 k-1
L (t) t L (t) k- L (t) k 2.
k k k-1 k k-2 '


The following result due to Saw and Chow (1966) connects the terms of a binomial series with the Tchebycheff-Hermite and Legendre polynomials. For any p,


m
(2.1.6) ( pr(1-p)mr c (r,min) = m (k) L (2p-1).
\r (m-k)!(2k). k
r=O


The importance of this result to the next section justifies the inclusion of the following proof.

After substituting ck(rm), as defined by (2.1.1), into (m-k) :(2k)'
equation (2.1.6) and multiplying the equation by (m-k)(2k) the
(k )2
equation to be verified simplifies to

m k
j ) (2k-j)Y(m-k+j)! r m-r
S(_)j () ( )k j!(k-j) p (1-p) =m!L k(2p-1).
r/- j ( -j .k

Notice that rj equals zero unless r > k j. Thus letting Q represent the left-hand side of the above equation and changing the order of summation, we have


k m
Q = (-l)j (2k-j)'(m-k+j)! 7-' (m) kjr(1-p)m-r
j.(k-j). L i)Q Yr1)r
j=0 r=k-j


Letting r = r-k+j, the sum in brackets simplifies to







16




m-k+j

I m pr I+k-j (1Pm-k-j-r'
r =0 (r--k+j -r')r (k-j) m-k+j

=(kmn pk-Jj (m-+j) r/1 mk+j -r/

r =0


= Q k-j



Therefore Q reduces to

k
(i m!(2m-j)! k-j
Q (-lj j(k-j)! (k-j).
j=0


Since by definition



E k (2p-1) = I1-2s(2p-l)+sj
k k


it remains to show that




m! s Q = 1-2s(2p-l)+sI



Making the change of variable =k j the left-hand side of the

above equation becomes






AL-0 j=0 + Li+

However it can be shown that [1-2s(2p-l)+s 2 also reduces to the

above sum.. We have







17







2

= (its) [i 4s 2

( 1( 1 + s )
-1 (-'%, (4p)


(1+s) E (-1

2 L (2+)! But 1\)may be written as (-I) I (,)
(2 41


since

2 2/ 21

2:


A 1.3 ... (22-1)
2



2 A! 2.4 ... 22

1 (21)! 4 A(A! )2


Substituting the above result into the expression for L1-2s(2p-l)+s 2]we have


[1-2s(2p-1)+s 71_ (2()! p
1=0( '


= O (2A)! (sp j -) (22+j\ ij

1=0 (A:)2 j=0


_~ (1j (2Ltj)! I j+),

2=0 j=0 Y I!

which completes the proof.







18




2.2 The Fundamenta! Result


In this section we use the results of the last section and Lemma 1, page 11, to obtain a finite seriies expression fori P r:m(p).
r:m

First we let p = 1 G(9') in equation (2.1.6). Then we have


Mi\ r m-r
E (r)[1-G(y) ]r[G(Gy)] m-ck(r,m)
r=0

2
m.'(k!.)
(m-k)! (2k) L [1-2G(ey)] (m-k)!(2k):' k


Multiplication of the above equation by g(y) and integration with respect to y gives

am

(2.2.1) E [1-G(9y)]r[G(ey)]m-r ck(r,m) g(y)
-= r=O

.2
= m!(k!) 2 Lk[1-2G(Gy)] g(y) dy .
(m-k)!(2k). k


Now taking the left-hand integral inside the summation and defining

Jk(p) by


(2.2.2) Jk(p) = Lk[1-2G(Gy)]g(y) dy,


2 -1
where p = (1+8 ) equation (2.2.1) becomes


m o
P r m-r
E ck(r,m) r [1-G(y)]r[G(y)] m-rg(y) dy
k=O

2
m!(k!)
= (m-k)!(2k)' k)







19



Finally, applying Lemma 1 to the left-hand side of the above equation, we have


m 2
(2.2.3) Ec (r,m) P (0) (m-k) (2k) p)
0 k r:m (m-k)!(2k). k


An alternate expression for Pr:m(p) can be obtained by noting that for fixed m the set of points Pr:m(p), r = 0,1,...,m, lie on a polynomial of degree at most m. Hence, for some constants, say e ,e ,...,e we can write
1

m
(2.2.4) P (o) = E e.c.(r,m).
r:m j=0 J 3


Multiplying (2.2.4) by ck(r,m) and summing over r yields


m m m
(2.2.5) E c (rm) P (p) = Z e.c (r,m)c (r,m).
r0k r:m r=j- 3 k
r=O r=O j=O

Using the properties (2.1.3) and (2.1.4) of ck(r,m), the right-hand
k
side above reduces to


m 2
ek E ck(r,m)
r=O

(k!)2 (m+k+l)
2k+1 /
e k /2k)
k /

Hence, the constants eoel,..., em can be determined by equating the right-hand sides of equations (2.2.3) and (2.2.5). That is,

(k!)2 (I'm+k+l) 2
\ 2k+1 m (k!)
ek 2k) (m-k)!(2k)! k)
\k.







20




After slight simplification, we have


e = r- m!2kl i (P), k =0,1,... ,m
kk L) (m+kl).


Letting b k (in) denote the constant in brackets and substituting into equation (2.2.4), we have


m
(2.2.6) Prm()= Z c k (r ,m) b k (M)in) )
k-_0


Before investigating the integral J () we should comment on the utility of the expression for P rm(p) given by (2.2.6.). Most important, by defining J k(p) appropriately, the expression is valid for both positive and negative values of rho. Next, the integralJk() does not depend on r or m. Hence, for a given value of p, only one set of values, J () k = 0,1,..., is needed. Also, as will be shown

in the next section, J p = 0 for odd k, thus decreasing the number of terms in the series by one-half. Furthermore, for large m, the series may be truncated without serious effect, since the factor c k(r ,m) b k(in) approaches zero as k increases.




2.3 The IntegralJk(P



Consider the integral



Sk()= Sf L k[l-2G(9y)] g(y) dy



defined for k = 0,1,... in the last section. Using the recurrence relation (2.1.3), it can be seen that the Legendre polynomial L kt) is







21



an even or odd function in t, doending on whether k is even or odd, respectively. Also, when rho is positive, G(ey) is the normal distribution function which implies l-2G(ey) is an odd function in y. Therefore, Lk[l-2G(ey)] is an odd function in y when k is odd and an even function in y when k is even. Since g(y), the normal density function, is an even function in y, it follows that J k(p), rho positive. equals zero when k is odd, and for even k



J k(p) = 2 S Lk[2G(ey)-l1 g(y) dy.
0


After making the transformation u = iy, J kesp) becomes


2







the distribution function G(u), we have


k(P) 2h Lk[2G(u)-I] e J dG(u)
2 2T





e 1

where p and O

O.
eI e +7


Next consider the integral J k(p) when rho is negative. Now

e
e is imaginary and is written as e = Zy where C. =



Hence, the function G(ey) appearing in Jk(p) is complex and in order








22




to simplify J p the real and imaginary parts of GCjcpy) must be isolated. Denote these real and imaginary parts by a'(cpy) and (Cpy), respectively. Then GWiyy) and its conjugate can be3 written


G (Lcy) = ;Cy) + i (cpy) (2.3.1)

G(-4ySy) = o(CPy) i a(yy).


Using the definition of GCpy) given in equation (1.3.2), we have





2ce((pyf = ~ e"t g(t)dt + f g(t)dt}



t CO




o(y)= 1Z -tp ~)d








Subtracting equations (2.3.1) gives


-2 2 0iy P
2i.F(9py) = er Oe g(t)dt J e g(t)dt}



-!e__22 f (etCYY e-iCY) g(t)dt.

0

it~py -itcpy
Since e -e 2j. sin (t~oy) we have



(2.3.2) 5(ePY) = e" YJI sin (tcpy) g(t)dt.
0






23



(Cpy) can be further simplified through integration by parts and differentiation of (2.3.2). First, integrating by parts,we have

2 2 CD
(2.3.3) $(py) = e y Y 1 cos (tpy)g(t)

COO

t1 cos (ty)g(t)dt
T_ 0


2 2 1 2
1 e 1_ y t cos (t~py)g(t)dt.
Tpy
9Y gg 9 o

Next, differentiating S(cpy) with respect to py, yields

d 2 2 m
(2.3.4) d (yy) e= y e sin (tpy)g(t)dt
0
2 2m

+ e 2Y ft cos (tpy)g(t)dt,
o


where differentiation was permitted inside the integral, since it cos (t~py)g(t)I < tg(t) which is integrable. Combining equations (2.3.2), (2.3.3), and (2.3.4), we have

dB(yy) 1 29 y2
d(cpy) = py (yy) + --2- gy $(Ty)
d(cpy)

2 2
ez


It follows, since $(0) = 0, that

-_ e dt.
PY f 2
(2.3.4) 8(Ty) = 1e dt.
o ,2i


Hence the complex function G(iy) can be written as






24




1 CP1;I ~2
G(19y) + e dt


1
= -+ iL(Cy)
2


Returning to the integral Jk(p) we can now write



Jk(p) = Lk[-2i$(cpy)] g(y) dy.


Although the Legendre polynomial has an imaginary component, its definition and recurrence relation still hold. In fact, Lk [-2L8(yy)] is an even function in y when k is even and an odd function in y when k is odd. Therefore, as in the case when rho is positive, Jk(p) equals zero for odd k and for k even



Jk(p) = 2 Lk[2i5(9y)] g(y) dy.
o
0

1 1
After making the transformations u = y and h = 2
e2 2
Jk(p) becomes

,-h-1

Jk(P) = 2(-h) L Lk(2i 5(u)] e-+2 dG(u),
kk Le'

1 1

where p and < p <0 imply that h<-m, m 2.
The following lema suarizes the results of this chapter.-1
The following lemma summarizes the results of this chapter.






25




Lemma 2

m
p r:m(P) = ck (r,m) bk(mn) J(),
k=0
k even where

()3 k (k! )3 k ij r (2k-j (m-k+j
c (r,m) (2k) (-1) .-j k j
k (2k)1 .= -j/ k2
j=0

m! (2k+l)!
bk(m) =
(k!) (m+k+i)! and for even k, Jk(p) is the integral


2 1 dG(u)
k(p= 2 fh Lk[h(u)] [e- u -dG(u)

with

1
h+l

and

2G(u) 1 h > O

h(u) =

21 u 2
2i I et dt h < -m, m 2 2


Lk(t) is defined by the recurrence relation

L (t) = 1
0
L Ct) = t
L1

2k-i k-i
L (t) tL (t) L (t), k > 2.
k k k-i k k-2
















CHAPTER 3


NUMERICAL RESULTS




3.1 Exact Results


In general the value of the integral J k(p) can only be

approximated so that exact results for P m(p) are not available. However, exact values of J (p) and J2 (p) can be found. Then, since
O2

Jk(p) is independent of m and r, P 2(p), r = 0,1,2, and P (p),
r:2 r:3

r = 0,1,2,3, can be determined.

Since L (t) -1 we have from Lemma 2, page 25,
0



J (P) = :2hl* 1L Ch(u)e dG(u)



hI e 2 du


= 1


J2(p) can be determined indirectly by first finding P2(p). Letting

m = r = 2 in Lemrna 2.

P 2:2 (P= P2(p) = C (2,2) b (2) J (p)



+ c2(2,2) b2(2) J2(P), so that

P 2p) c (2,2) b (2) (31.) J p =2o o
2(P) c2(2,2) b 2(2)


26







27




The value of P2(p) can be found in closed form by integrating the original expression for P m(p) given in equation (1.2.8). With m = r = 2, we have 22
x1+X2 2px xx2
2
1 2 -~ 2(1-p dxdX
P2(p) = (2) (i-p ) e dx12dx2
0 O


Making the transformation


x1-p x2
u1 (1_p2)


u2 = x2 it follows that


x1 = (1-p2) u1 + pu2


x2 = u2,


and the Jacobian of the transformation, J(x,x ulu ), is




J(x1,x2 -6 ul,u2) =
01


= (1-p 2 Therefore, r1 2
P ) = 2 e du du
2 J i 21 2 1
o (1-p )U
p l1







28




Finally, making the polar transformation,


u = r cos 9


u2 = r sin 9,


we have

-1
cos (-p) 2
P2 = T dG r-e dr


1 -1
2 cos (-p ) 2T
1 (,2 -1
S + sin p)


1 1 s.n-1
=- + sin p.
4 2TT

Next we need the values of c k(r,m) and bk(m) for k = 0 and

k = 2. For k = 0, c (r,m) = 1 and for k = 2,
0

1
c2(r,m) = r(r-1) r(m-1) + 1 m(m-1).
2 6

1 2 1
Thus for m = 2, c (r,m) equals 3' 3, and 3 for r = 0,1, and 2,
2' 3 3' 3

respectively, and for m = 3, c2(r,m) equals 1, -1, -1, and 1 for

r = 0,1,2, and 3, respectively. The constants b O(m) and b (m) are
0 2

given by
1
b (m) ,
o m+l

30
b (m) =3
2 (m+) (m+2)(m+3)


Substituting m = 2 and m = 3 gives

1 1
b (2) b (3) =- ,
o 3' o 4

1 1
b2(2) = b2(3) = .
2 2'2 4







29




Now using (3.1.1) we find that

3.-i 1
J )(=P sin p
J2- T 2


Finally, substituting the above results into Lemma 2, we have, for m=2,

1 1 -1 :2 = 4 T T p

1 1 -1 1:2 2 17

1 1 -1
2:2( = + sin p and for m=3,

Po:(p) p i
0:3 8 4T

3 3 -1
Pl:3p) 8 4- sin

3 3 .-i P2 3(p) -j sin p

:8 47 sin

1 3 .-1 (P + sin p.


Notice in the above results that


r: mm-r:m( and
m
E p re(p)= .
r=O

These properties also hold in general. The first result follows immediately by letting r= m-r in Lemma 1, page 11, and noting that G(Oy) = 1-G(-ey) for both positive and negative rho. Lemma 2 is used
m
in showing the second property. Since Z c (r,m)=0 for k= 1,2,..., r=O k
we have






30




m m m
E P (p) = Z Z b (m) c (rm) J (p)
r:m k k K
r=O r=O k=0

m
= b (m) c (r,m) J (p) r=0O

m
1
m+1
r=0

= 1.
1
P r:m(p) can also be computed exactly in the case when p = .
r:m

From Lemma 1,


P (1) (m) [1-G(y)]r [G(y)]m-r g(y)dy,
r:m 2 r

1
since = i when p = But

{m rm-r()
(m [1-G(y)]r [G(y)] g(y)



1 r (m+1)! m-r r
m+-1 m [G(y) [1-G(y) g(y) ,
m+1 (m-r)!r.


where the term inside the braces is the density on the (m-r+ 1)st

order statistic from a normal random sample of size (m+ 1).

Therefore,

p 1 1
P () r = O,1,...,m.
r:m m+1

This last result implies that Jk( ) 0, k2. From Lemma 2,



Pr:m( ) = Z bk(m) ck(rm) J( 1)
r:m2 k=0 k k k
k--0
k even

1 m
m- + E bk(m) ck(rm) Jk2 k=2
k even







31




Since the last sum equals zero for all r and m and since b k(m) c k(rm) does not equal zero, J (-) =0.
k 2



3.2 Evaluating the Integral J k(P)


The Case When Rho Is Positive
-1
Recall from Lemma 2, with p = (h+l) > 0, that



SkL(p 2 -h L]k [2G(u)-] I eh- dG(u)



Before attempting numerical integration, the integrand should first be investigated for different values of k and p. For given values of G(u), the Legendre polynomials can be evaluated from the recurrence relation (2.1.5). Also, the value of u corresponding to G(u) is given to eight decimal places in The Kelley Statistical Tables. For each of
1 1 9
the cases p = T0, and 9 the integrand is plotted in Figures 1, 2, and 3, respectively, for both k= 2 and k= 10.

Since numerical integration is most accurate when the function being integrated is well behaved, we should expect good results for
1
p< -< with the accuracy increasing as rho decreases. Also, since k is the order of the polynomial being integrated, the better results
1
should occur when k is small. When p >, the integrand approaches infinity as G(u) approaches one as can be seen from Figure 3. In this case, it is not likely that numerical integration will give accurate results.

Using Simpson's rule with 200 intervals, the interval width
.5
equals .0025, with G(u) taking the values .5 + (.0025)j,







32





1. 0 3 Ll,42 G~ 1 [e2 -,









.5











k 2
h 1




-10




/-1.0


1=I
Fir 1. Tefnto h2 k2 u)-1 2
I /0







33




1.0~. -i I23 ) 1









5I



























-1. L-1 k2
k 10










-1.h



Figure 2. The function h2 LK,[2G,\u' 11 Le-21J

p =1/2.







34


1.5 ~.L. [2 G(u) -1I[eu Ie


















0 G1u
0






k5 2












9-
Figue 3 ThGucto 2Lu)~)-1 e





k = 910.







35




j =0,1,...,.200. Then J k(p) can be approximated by
K2

j.0025 200 i .h-1
Jk (p 2h' -32 c L (.005j) e
k ~ j=o


where

C =C =1
o 200

4 j odd,


S,2 even,

and u is the value of u satisfying G(u) = .5 + (.0025)j.


The Case When Rho Is Negative

With p <0, the Legendre polynomial has the imaginary argument if(u), where f(u) is given by u i 2
f(u) = 2 1, eit dt.
V o


For given values of u, the function f(u) can be evaluated quite rapidly by first expanding the integrand in a Maclaurin series and integrating term by term. That is,

f~) 2 U O t 2j
dt
f(u) = 2 d
,/7 o j=O 2 j.


2 CD 2
t J dt
,rT j=O 0 2j!


2 u2j+1
j =0 /21 21 (2j +1) j.







36




Letting f.(u) denote the (j-4l)st term of the sum, we have the recurrence

relation

2
f (u) u


2
f(u) u 2j-l f u). j = 1,2,...,
j 2 j(2j+l) j-1

which can be easily programmed.

Although the Legendre polynomial has the imaginary argument,

if(u), for even values of k it is a real function. Hence, for computing purposes, we can avoid complex numbers by defining the function
.
L (-t) by the recurrence relation,
k

L *(it) = 1
0 .
(3.2.1) L (it) = -t
1
2k-l t k .~t k-i ( ), k>
Lk(-t) = (-1) k- t -- (it) k 2.


Then the function Lk (it) can be determined by


L (it) k even
L(it) = k t
-i.Lk(.t) k odd.



The above relation can be verified by substituting into the recurrence relation (3.2.1) and comparing the results with the recurrence relation

for L k(t) given by (2.1.5).

Once the Legendre function, L k[if(u)], is evaluated, we can
2-1-h-1
examine the integrand, (-h)* Lk[Zf(u)] Le J of Jk(p). The

integrand is plotted in Figure '4 for p = -7' k = 2, and for
1
p = 1 k=2 and k= 10. Notice that the scale of the graph differs

from those in Figures 1, 2, and 3, since,for large k,IJk(p)I is quite








37


5 (-11)2 L [jif(u)] Ie2'
k L J








0- - --






















-150 i?











p=1/2, k-2 p 1/10, k 2






-2 Fig- /1 ure 4. The function (-h)2 I1 [if (u)J [e 2

p --1/2, p -110






38




large. Also, unlike the case when rho is positive, the functions do not cross the G(u) axis. These differences, together with the smoothness of the curves, should make numerical integration even more accurate for p < 0.



3.3 Computing Pm(P)



Once the J (p) k = 0, 2,... ,m, are computed, P (p) is deterk .r: m

mined, for r = 0,1,... ,m, by the expression given in Lemma 2. That is,


m
(3.3.1) Pr (p) = Z bk(in) ck(r'm) Jk(p)
r:m k k k
k=0
k even


Unfortunately, since the Tchebycheff-Hermite polyrnomial, c k(r,m), cannot be expressed in a recurrence relation, it is difficult to compute

Pr m(p) using the above expression, except for small values of m. However, with r=m, c k(r,m) simplifies to. say a (m), m!. (k!)
(3.3.2) c (i) = m- (k)


Furthermore, the probability P m(p) can be written as a linear combir: i

nation of P k(p), k = r,r+l,... ,m. That is, m-r
(3.3.3) Pr: m(P) = (r 7(-1) ( P r+j (P'
j=o


(The proofs of (3.3.2) and (3.3.3) are given at the end of this section.) Hence, we use (3.3.1) only in the special case when r=m. Relation (3.3.3) then can be used for computing P m(p ) when r i m.
r: i







39



A further simplification in the computation of P (p) can be
m

made by combining the constants bk(m) and ck(m). Letting dk(m) denote the product, we have


dk(m) = bk(m) ck(m)

2
m!(2k+l)! m!(k)

(k:) 2 (m+k+l)! (m-k)! (2k)!

(m.)2 (2k+l) (m+k+l)! (m-k)!


The constant dk(m) can be computed for even values of k by the recurk
rence relation,

1
d (m) =
o m+l
(3.3.4)
2k+l m-k+2 m-k+l d (m) = d (m), k = 2,4,... ,m.
k 2k-3 mik+1 m+k k-2


Finally, combining the above results, we have the following computing formula for P (p):
m
m
(3.3.5) Pm(p) = E dk(m) Jk(P
k=O
k even


The proofs of (3.3.2) and (3.3.3) follow. To prove (3.3.2), we first let r=m in the definition of ck(r,m) given by (2.2.1). Then, writing combinations as factorials and cancelling like terms, we have


ck(m) (ki k)3m ) 2k-j m-k+j
k (2k)! 1 j=O ) k j
3=0

(k!)2 m! k(2k-j)
(2k)! (=!k-j) '.







40



Thus, we must show that the summation above equals one. This summation can be written as

k
j 2k-j\ k1
E (_) (k / (k-ji
j=0


Then, letting Z = k-j, the result to be proved becomes k-1 k+1) (k)
(-1) = 1.
=0


Next, we introduce the negative binomial and binomial identities given by


1 r (k+r~ r
(3.3.6) k+l (_-1) (kr ar
(l+a) r=0


k k (3.3.7) 1 + = Z k



Multiplying the left-hand sides of (3.3.6) and (3.3.7),gives


k
(l+a) 1 1 ( L
k k+l k k (-) a
a (1+a) a (1+a) a k=0


Equating this product to the product of the right-hand sides of (3.3.6)

and (3.3.7), we have


1 k r k+r k rSE (-1) a = E (-)1 ) a
a 2=0 r=0 =0

0 k
Finally, equating coefficients of a and dividing by (-1)k, we have

k I-k k+ k\
E (-1) = 1,
a=0

as was to be shown.







41



In proving (3.3.3), we begin with the basic definition of Pr:m(p). That is, (mr
P (p) =() PX >0 ...'Xr >0; Xr+ <0,... X <0.
r:m r \ 1 r r+1 m

Next, let A. be the event [X.>O] and denote its complement by A.. Also, define I as the intersection A1A ...Ak. Then Pr:m (p) can
k 1 2 k r:m
be written as


P (p)= P ...A
r:m r 1 r r+1.. Xm



w/ r 1A.. r Ar+1- XM)


= ( P(I r) -P(A,.. .AA ..A .
(r r1r r+1 -m)

Applying de Morgan's rules, and since P(I ) = P (p), we have r r
(m \r- m-r
r:m ( (r) LPr(P \P( 1- Ar U Ar+j
j=1


( ) r(p)PCPUA..AAr. .
r (P L r= 1- r r+j
j=1

Finally, using the formula for the probability of a union, we have the desired result. That is,

(m{ em-r

P (0) = Pr (p) r P A AA
r:m r r L 1 r r+j)
j=1

Z P(A ...AA A )
+1r r+3 r+A
1

+ E E P(A...AA *A A
J < j<" r r+J1 r+J2 r+j3







42



F (A Arr A





S+(-r)(mr)



,-r ) -m /m-r 1 (I
3 (Ir+3) +..-+ _m -r


(M) m _)j(m-r P (P
rj=0 )rj




3.4 Accuracy of the Results


The integrals Jk(p), the constants dk(m), and the probabilities P (p) were evaluated with double-precision accuracy using an IBM model


360 computer. The computed values of Jk(p) were checked against the
1
exact values for k=0 and k=2 and for p =. For k=0 and k=2.

the results were accurate to at least seven significant digits for !1 I and to five significant digits for 1 1 was


computed accurately to six significant digits for k< 14 and to five
aIf 1
significant digits for 16< k < 22. Hence, for 1p we would

expect Jk(p), k>2, to be accurate to at least the fifth significant digit. The computed values of Jk(p), k< 22, are given in Appendix 1
1 1
for p = -, p = 2(1)25, and for p- p = 3(1)26.
p P
1
As expected, with p > ., accurate values of Jk(p) were not obtained using the method of quadrature described in Section 3.2. Further investigation has to be made in order to find a means of

evaluating Jk(p) accurately for p > .







43




The constants d k(m) were evaluated exactly using the recurrence relation (3.3.4). They are tabulated in Appendix 2 for m<25, k = 0,2,.. m.

Finally, the probabilities P (p) were evaluated using the
m

formula (3.3.5). Results for p >0 were compared with Ruben's tables and were found accurate to at least five decimal places. For p <0, Steck's relation,



MI) P (1:2) mm 2 k PRl(k -) P-k Y '
rn 2 -k=2 LI

k even


was used in making comparisons. Again, the computed values of Pm (p) were accurate to at least five decimal places. P (p) is tabulated in
1
Appendix 3 for p = -, p = 2(l)(25) and 2 < m < 22, and for
1
p p, p = 2(1)21, and 2 < m < p.
















CHAPTER 4



APPLICATION: A TEST FOR NORM-LITY




4.1 Introduction


Consider using a random sample, Y 1 Y 2 ...,Y n from a continuous distribution F to test the hypotheses: H F is a normal distribution
0

H F is a skewed distribution.
a

As David (1962) suggested, one might consider the number of observations larger than the sample mean as a test statistic. Letting


1 if Y. >
i

0 if Y < V for j = 1,2,...,n, the test statistic, Y, can be expressed as


n
Y = E q i .
j=1


Without loss of generality, we can assume that the variables in the sample have been standardized to have zero mean and unit variance. Then, using the representation of Y given by (4.1.1) and assuming that H 0 is true, we can find the mean and variance of Y. For the mean, we have





44







45




[n
E(Y) ELZ C j=1

n
= Z2 E (Cp) j=1

n
= P( = 1)
j=l

n
= P(Y Y > O)
j=1

n
1
2
j=1

n
2'


since Y. Y is symmetrically distributed about zero. Before

calculating the variance of Y, we first need the variance of CP and

covariance of c; and k. We have


Var (p.) = E(.) [E(j)]


E (C j) ( -)2
2

1


and

cov (jk) = E(p k) E(c ) E(Cpk)

= P(Y -Y>O, Y -Y>O) Sk 4

But the probability above is identical to P2(p), where p equals

cov (Y.-Y, Y -Y) corr (Y. -Y, Y Y) = k
3 k
A/Var (Y.-Y) Var (Y -Y)
3







46




The covariance of Y.- and Y k- Y and variance of Y. -Y are


coy (V IY k) -coy (Y .,Y) -coy (i-Yk Y+cov (YY)


0 1 1 +1
n n fl


n

and

Var (Y- 2 cov (Y.,Y + Var(Y


2 1
n n


n

respectively. Hence,


1


n





Therefore,


co (%Jk~ =2 (n 4i)


1 si N and

(jn
Var (Y) =Var (z )


n
Z Var (cp)+ 2Z cov(CIT, k



n +n(n-1) si-1 (-I ) T 2-r s 1~







47




The standard deviation has been computed for n< 50 and is given in Table 1.



4.2 The Null Distribution Clearly the test statistic, Y, has a discrete distribution,

taking on values 1,2,...,n-1 with positive probabilities. The probability of the event, [Y=r], r = 1,2,...,n-l, can be written as


P (Y=r) = P(Cpi = 1,. ..ir = ;ir+ = 0'... 'i =0) ,
n L"1 " r+ 1 n



where the summation is over all partitions fi ,...,i ; i +1,....i n 1 r r+l n
of the set [1,2,...,n]. Then, since the variables 11'2 ... Yn are

identically distributed,


P (Y=r) = r P(1 = 1..... r= 1; Y r+=0,... n=0)
n\r /1 'r' r+1' n 0

= (n)P(Yl-Y>O,... ,Y-Y>0; Y r+ 7-Y<0,
(r 1r r+1

...,Yn-f<0).

After the transformation


U. = (Y. Y) n
Jrn
3 3 \n-17
we have


P (Y= r) (n) P(U >0,... ,U >0; U <0,... ,U <0),
n r/ 1 r r+l1 n

where the normal variables U 1,U2 ...U n, each have mean zero and common variance and correlation given by







48








TABLE 1

STANDARD DEVIATION OF Y


n n

3 .50000 27 1.56581

4 .59242 28 1.59456

5 .66760 29 1.62281

6 73389 30 1.65058

7 .79416 31 1.67788

8 .84994 32 1.70475
9 .90214 33 1.73119

10 .95140 34 1.75724

11 .99818 35 1.78291
12 1.04283 36 1.80822
13 1.08562 37 1.83317
14 1.12678 38 1.85779

15 1.16646 39 1.88208
16 1.20484 40 1.90607
1.24202 41 1.92976

18 1.27811 42 1.95316

19 1.31320 43 1.97628

20 1.34738 44 1.99914

21 1.38071 45 2.021-93

22 1.41325 46 2.04408

23 1.44505 47 2.06619

24 1.47617 48 2.08806

25 1.50664 49 2.19701

26 1.53651 50 2.13112







49




Var (U.) n L Var (.Y,
3 n-i


n (1 1 )
n-i n




and

corr (U.,U n .- coy (y, -Y, Y -Y)
k n-i

n (_1
n-i n

1
n-i


Hence, for m
Ult 2 .... ,U., have a multivariate normal distribution. Therefore,


P pk < m, is defined and can be computed, using the method discussed in Chapter 3. Furthermore, since P(Y=rn) = 0, we can set P (p) equai
n

to zero and use relation (3.3.3) to find P n(Y= r). That is, (4.2.1) Pn(Y= r) = (n n r (-) j (Y-) p




where P -1 and P n(p) = 0. Using equation (4.2.i) and

the results of Chapter 3, the null distribution on Y was obtained. The results, for n < 22, are given in Appendix 4.




4.3 Approximations to the Null Distribution


David (1962) has shown that the asymptotic distribution of

1 i
is normal wihmean zeoand variance
wit zeo- Hence, for
n 4 2TT

large n, we should be able to approximate the distribution on Y, using a normal distribution function. In particular, the critical values of








50




Y needed to form the rejection region can be determined using the approximation. With a 10%1 level of significance and a two-tailed alternative, the critical values are the solutions to the equation

1 n

2 2 = 1.645

vVar (Y)


(Notice that 1. was added as a correction for continuity factor.) As an example, with n = 19, we have 1 19

1.31320 = 164

or

r =9 2.1

a7, 11.

As a check, from the small sample distribution on Y given in Appendix 4, we have

P 1 (Y< 7) = P 1 (Y;>ll) = .0595.


Table 2 compares the small sample distribution of Y with the normally

approximated distribution for n= 19. (The results of the approxiriation which will be discussed next arelisted in the third column.) We would expect the approximated results to increase in accuracy as the sample size increases.

An alternate approach to the problem of approximating the null distribution on Y is through the use of order statistics. Consider again the random sample of standardized variables, YY 20 ... Y.n Then, letting Y 0)denote the Vth largest order statistic, the events [Y< r] and CY(v :7 ] are equivalent if we set V equal to n -r. Letting F (x)







51




TABLE 2

THE CUMULATIVE DISTRIBUTION OF Y, n= 19 Approximations

Small
Sample Edgeworth' st
r Distribution Normal Expansion


1 .0000 .0000 .0000

2 .0000 .0000 .0000

3 .0000 .0000 .0002

4 .0000 .0000 .0007

5 .0007 .0011 .0029

6 .0092 .0112 .0142

7 .0595 .0639 .0638

8 .2190 .2231 .2175

9 .5000 .5000 .5000

10 .7810 .7769 .7825

11 .9405 .9361 .9362

12 .9908 .9889 .9858

13 .9993 .9989 .9971

14 1. 0000 1.0000 .9993

15 1.0000 1.0000 .9998

16 1.0000 1.0000 1.0000

17 1.0000 1.0000 1.0000

18 1.0000 1.0000 1.0000


Uses f irst two Troments of Y.
t
Uses first four moments of Y ( -Y.







52




represent the standardized distribution function of Y(- Y, we can write

(4.3.1) P (Y < r) P(Y O)


PY( )-Y- <


2 V 2 v







where aIVnd 2 represent the mean and variance of Y(respectively. An approximation to the distribution F V(x) can be obtained by using Edgeworth's expansion (see, for example, Cramer, p. 229). Letting kgv represent the kth central moment of Y(V)7, we have, using the first four moments, (4.3.2) F (x) L G(x) g(x) 1 3 %) (X2

(4.3.2 F6 3/2(x-)



13
+ -4 3) (x3 3x)
2 V



where G(x) and g(x) are the standard normal distribution and density functions, respectively. Thus letting x in equation (4.3.2),

2 v
we can approximate P (Y r).
n

By using a power series representation, the moments, k k = 2,3,4, can be determined. Saw (1958) has shown that the kth moment, k%, of the Vth order statistic can be expressed as







53



1
(4.3.3) k4V = H (p ,O,k)
j=0 (n+2)


= E H.(pv,k) n(j), j=0


where p and, for convenience, we have replaced H. (p ,0,k) and
V n+1 j v

-1 by H (p ,k) and n(j), respectively. The constants, H (p ,k), (n+2)j V
are tabulated for k = 1,2,3,4,j SE 5, and p = .50, .55,...,.95 by Flora (1965). It follows, since the moments kE are functions of kV' that


k = C.(p ,k) n(j),
j=0


where the constants C.(p ,k) are functions of H (p ,k).

In order to find the C,(p ,k), we must first express the .3 V

k Vas functions of the kP Letting Y(t) represent the characteristic function of Y -Y, we have



cp(t) = E[e ( )]

Since Y and Y )-Y are independent, we can write



_(t) = E [e t(Y)
StY









2n (it)j
.e



E=e

j=0







54




00 it () 1 O

1.0AL(2n) jz


C O 2 O 2 1 + j i t

1=0 j=O (n


After the change of variable, k = 22 + j, it follows that


~~(t o (it)Jk Fk/2 k!______k=O L1=o 2(k-21)! (2n) (-2V


Therefore,


-k k/2 k

2!(k-22e)! (2n)2 and, in particular,


E(Y 0- Y) =IP


E(Y -) =1
00) 2 V n

E( y3 2
(V) Y 3PV n 1V


E (Y 46 2
\O- Y) 4= I -n 2 .\ +
n


Using this information, the central moments, k of Y 0- Y can be

expressed as functions of k4V For example, with k =2,


2 (YCV- 2
~ E(Y 2-Y- 4+

2 2 V) n V 1

-22



2 2? 'V -1PV n







55



Similarly;


V 3 \ 3 2N "V +p 21

6 2 3 _462
4 V = 4P ~V + -2 1~ 3 4V 1 6 YV 2V
n

6 3P4
-n 2PJ 1- I


Finally, the constants, C.i(p.,k), can be determined as functions of H (pvk) by substituting the power series representation of k P and

equating coefficients of n(j). With k = 2, we have


2 -1
2 v 2 2V -1V- n


=Z H.(p\,,2) n(j) H (p,1) n~j
j=0 J L iv=0 '


n:72 2'
1 n+ 2



E H.(p,,2) n(j) E H.(p 1) n(2j)
j=0 J =


-2 E E H (p 1) H (P ,1) n(j+k) Z 2 2' n(j+1)




E C~ i (P,2 n(j)


Equating coefficients of n(j), we find that


C (pV2) = Ho(p2) -H 2(pV,1),


C 1(p ,2) = H 1(pV2) H (pV1) H 1(pV 4) -1,



2 v' -2 1 V'op~)H(v 2 ,







56






2H 1(PJI) H 2(Pvtl) 4, C3 (P ,2) = H (p 2) 2H (p 1) H (pV,1)


3 %02 o 2 3 0V 4
2H (p ,1) H (p 1) 4,

2

C4 (P,2) = H4 (PV,2) H2(p ,1) 2Ho(Pl) H4(PV,1)


2H(PV,1) H3 (PV,1) 8


C (p ,2) = H (P,2) 2H (p ,1) H5(p ,1)
5 V) 5 0' o V 5 IV

-2H (p,1) H4 (PIl) -2H2 (pV,1) H3 (VP,1) 16.


The constants, C (p ,3) and C (pu,4),can be found in the same tedious manner. The values of C (p ,k), k = 2,3,4, j' 5, and pV = .50, .55,..... 90, are tabulated in Tables 3, 4, and 5.

Using these tables, we can approximate the moments 2%) 3 V' 1p9
and 4%, and then use relation (4.3.2), with x = 1 in order

2 V

to obtain an approximate distribution on Y. As an example, we take n = 19. Then


p
V- n+l

19-r
20


Tables 3, 4, and 5 were used to approximate 2f' 3f and 4, for r = 1,2,...,9. The results are listed in Table 6. The values for 1P were taken from tables computed by Teichroew (1956). Teichroew's tables were also used to check the accuracy of the series approximation for 2~ For n = 19, the approximation was accurate to five decimal places.














TABLE 3

c (pv 2)




p \Vl 0 1 2 3 4 5

.50 Zero .57079633 .46740110 -.53726893 -3.86976182 -12.0889819

.55 Zero .57983932 .49410800 -.48836662 -3.82833586 -12.1396154

.60 Zero .60792651 .57852616 -.33227941 -3.70150569 -12.3352921

.65 Zero .65821859 .73530568 -.03640528 -3.48374825 -12.8518246

.70 Zero .73711417 .99616602 .47260438 -3.17871571 -14.2029896

.75 Zero .85676747 1.42762451 1.35866689 -2.85548457 -18.0128221

.80 Zero 1.04137154 2.18241438 3.03241842 -2.93521545 -30.4943719

.85 Zero 1.34534536 3.67686183 6.76962950 -6.03082075 -84.1011123

.90 Zero 1.92211072 7.45692711 18.3890366,2 -35.47326470 -474.58344823





4














TABLE 4

c i (pv 0 3)



p \ 0 2 3 4 5

.50 Zero Zero Zero Zero Zero Zero

.55 Zero Zero .14261956 .46342873 .77469507 .2860125

.60 Zero Zero .30026404 .99445568 1.67177376 .4710749

.65 Zero Zero .49242777 1.68698946 2.86385938 .3195549

.70 Zero Zero .75034258 2.70841421 4.66613950 -.8926114

.75 Zero Zero 1.13308641 4.41851254 7.77768619 -5.7261556

.80 Zero Zero 1.77168120 7.74697987 14.08664124 -25.1106438

.85 Zero Zero 3.02054891 15.75323340 30.27071486 -122.3376979

.90 Zero Zero 6.18793059 43.35945955 93.78561212 -932.6502938






00















TABLE 5

c i (pv 0 4)



p \Vj 0 2 3 4 5

.50 Zero Zero .97862534 2.30187673 .68109061 -14.7500839

.55 Zero Zero 1.00864090 2.46469125 1.08317896 -21.1792945

.60 Zero Zero 1.10872392 3.01034124 2.48383168 -13.8178825

.65 Zero Zero 1.29975514 4.11498558 5.51922615 -11.4702010

.70 Zero Zero 1.63001191 6.20062897 11.82707278 -5.4443069

.75 Zero Zero 2.20215147 10.27259002 25.71608407 9.5845455

.80 Zero Zero 3.25336401 19.03426372 60.41653805 41.8041794

.85 Zero Zero 5.42986248 41.49665868 168.6658848 71.0033502

.90 Zero Zero 11.08343110 121.2957630 683.1599674 -891.8775674








60




These moments were used in the expression for F V x) given by (4.3.2) to obtain an approximate distribution on Y for n =19. The results are given i~n Table 2, page 51. For li= 19, there appears to be little difference in accuracy between the two approximating methods--certainly not enough to justify the extra labor involved in computations for the latter method. However, the second method does work well and is at least of theoretical interest.







61









TABLE 6

MOMENTS OF Y(V)- Y, n = 19


r 2J **4


1 18 1.37994 .11015 .01897 .04152

2 17 1. 09945 .07308 .00868 .01768

3 16 .88586 .05484 .00492 .00975

4 15 .70661 .04416 .00308 .00624

5 14 .54771 .03739 .00202 .00442

6 13 .40164 .03298 .00131 .00342

7 12 .26374 .03020 .00079 .00285

8 11 .13072 .02866 .00038 .00255

9 10 .00000 .02816 .00000 .00246



= n-r.

lv = E(Y ()- Y).

tk = E((V)- y k
i ) k =2,3,4.

















CHAPTER 5


OTHER METHODS OF EXPRESSING P (P)




5.1 Introduction


Before discovering the expression for P M(p) Involving

Tchebycheff-Hermite and Legendre polynomials, three other methods of expressing P m(p) were used in attempting to obtain numerical results. Each of the first two methods, outlined in Sections 5.2 and 5.3, expresses P M(p) as an infinite series. However, in each case, not only is the series slow to converge, but no workable expression can be given for the kth term of the series. Therefore, these methods are not useful in obtaining accurate numerical results. In Section 5.4, we give an expression for P m(p) involving the moments of extreme order statistics. However, this expression can be used only for limited values of m and rho.


5.2 A Power Series in Rho


Using the definition of P m(p) given in Chapter 1, we can write


P (P) = I... I' g(x ,... ,x ;p) dx . .dx
0 0

where g(x ,..., mx p) is the multivariate normal density on the equicorrelated variables XV,...PXm. Since, when rho equals zero, the density function simplifies to



62







63
m
1 2
m -- Zx
2 2j=1 g(xl,...x ; 0) = (2r) e
1 mi

m
we could simplify the integrand by expanding g(xI,...,xm; p) in a

Maclaurin series in rho. We have

O k k
(5.2.1) g(x1,... ,xm; P) = P g(x1,...,x ; P)
k=_0 P m p=0


Unfortunately, as a function of rho, g(x1,..., xm ; p) is quite complicated and it is not feasible to take derivatives with respect to rho.

However, the following identity simplifies the problem to.some extent:



(5.2.2) g(x ... ,xm; p)
p=O

6 6\ k ; EE Tx-) g(x1,...,xm; 0)
i
In proving identity (5.2.2), we use the characteristic function,

q(tl...',tm), of the variables X, ...Xm. By definition


m
LE t.X. (5.2.3) c(t ,...,t ) = Ee j=l

1 m
m
CO CO E t.x.
j=1 3
= ... e g(x ... ,x ; p)dx ...dx
1 1 m
_CO -00


-I t'Vt
= e

m
1 2
--- t p t.t
e 2 j=l J i






64



Differentiating with respect to rho, we have

m
t.x.

... e j=1 g(x1 ...,x ; p)dx1...dx


1 2
- t pE' tmt E~ t. 2~ t t
2 <. j k i = t.t. e
i
1
Since, for < p < 1, g(x ,...x ; p) is a continuous function
mn-1 1''m
8
in p, P g(x1,...x; p) exists and is integrable. Therefore, differentiation inside the integral is permitted, and, at the point p = 0,

we have
m
O CO E t.x.

(5.2.4) L g(x1,-. ,xm; p) e 6dx1...dxm

p=0
m
1 2
E t
=- 2 t.t. e i

Next, we consider the characteristic function of Xi,...,X when p =0.
1m From (5.2.3) we have

m
CO I E t.x.
j=1 ; ...J e g(xl,...,x ; O)dx .. dx
-_m _.. 1 xm 1 m


1m 2
--E t
e2 j= j
2j=1







65



Differentiation with respect to ta and tb gives

m
O m E t.x.
F P 6 6 j=1 33
... e g(x ...,xm; O)dx ...dx
-F a b

1 t2 8 6 2 j
=*e ot ot
a b

so that
m
CO CO Z t.x.
] j=1 iJ
(5.2.5) ... (-xaxb) e g(x1 ,...,xm; O)dx1...dx

m
j J xaxb1 m .1* m


1 m 2 SE t 2 jlJ = tt e j=1
ab


Again, differentiation inside the integral was permitted, since the

function xxb g(x ,... ,xm; 0) exists and is integrable. Since
at 1 m

6 6
xA g(x1,... ,xm; 0) -T -x-- g(x1... ,xm; 0), a b


after summing both sides of equation (5.2.5) over values of a and b

such that a < b, we have
m
O /.L Et.x.

(5.2.6) ... 0 g(x ... xm; 0) e J=1 dx1...dx
FO r E E 6X F g (x ....
-c a
m 2
2. j
1 =e 2 j1
a






66



Finally, addition of equations (5.2.4) and (5.2.6) gives



-m -m p =0
6 8"
a x & .'X -1
b1 E g(x1,... ,x;0) a
m
I E t.x.
j=1 a
*e dx1...dx =0,
1 m

which implies that

6 6 6
g(x1,..., x;P E (x1,..xm;O)
p=0 a

Since derivatives of all orders exist and are continuous, the preceding process can be repeated any number of times. Hence, the identity (5.2.2) is proven.

Using this identity in the Maclaurin expansion of g(x ... ,Xm;P),
1' m
given by (5.2.1), and substituting the resulting expression into Pm (p), we have



(5.2.7) PmP ...'J E k g(x1,...,xm;0)dx1...dx
o o a k=0



= * a k=0
k0 o o a < b a xb

The utility of this expression for Pm(p) depends on how readily each term in the series can be determined and on how quickly the series converges. Differentiation and integration are no problem once the







67




sum, I x-- has been expanded. In fact, the jth derivative
a of g(x ..., x ; 0) with respect to x is given by


g(xl ... ,x ; 0) = (a g(x.) g(x)
ix- 1 m \ida 1~ d jx x a
a a


= (-1)j H.(x ) g(x ...,x m; 0), j a 1 m

where H.(x) is the jth order Hermite polynomial in x. For example,
3

2
H (x) = 1, H2(x) = x 1,
o2
3
H1(x) =x, H3(x)= x 3x.


It follows that



(5.2.8) g(x) dx = g(x)J
O O



= (-1)j-1 H j1(x) g(x)J
O
0

H (0)
= (-1)



For even values of j, the (j-l)st order Hermite polynomial vanishes

at the point x=0. Therefore, when expanding the sum, we need to

consider terms that involve only derivatives of odd orders.
6
Denoting --- by 8a, we can represent the kth power of the
a
sum by the multinomial expansion







68




k k! k1 k2 k
E E 6 6 ) P
a



where N = (2) and where the summation is over all integer values of
N
klfk k satisfying Z k. = k and k. 0, j = 1,2,...,N.
V fj=l JJ '

Using this expansion and the value of the integral given by (5.2.8), we can determine the first few terms in the series expression (5.2.7) for Pm (p). Denoting the (k+l)st term in the series by S, for the first two terms, we have


0 M c
S g(x,...,x; 0) dx...dx
o o


= ()M

and


k1 .... k. 1 2 1 3m-1 m
o 0 1 N


g(x1 .... xm)dx1. .dx 1 1 m


N
Since E k. = 1, there are N = (2)terms in the sum. Therefore,
s1 J
since X,..Xm are identically distributed, it follows that







69




o o 9 (x l .. . x O )d ' d X m

o
o2 o 1 1- m




0
M'" H (0) 2 -2





(lm m (2)



where m(k) will be used to denote the permutation of m elements k at a time.
N
For the third term, S 2, Z k = 2. Since a value of k.
j=l

equal to two would result in an even powered derivative, we need to consider only values of k. equal to one. Furthermore, these two
3
"ones" must be assigned to two of the exponents in the quantity
kl(1 3k2.. k N

1 62 1 3) (6m1m) so that the resulting product contains
four distinct 5's. The first "one" can be assigned in N = (M) ways.
k.

Then, there remains (m 2 couples, 6i, containing 's dis1
tinct from the couple (6ii determined by the assignment of

the first "one." Hence, the second "one" can be assigned in (m 2) ways. Since the "ones" are not distinguishable, there are possibilities for selecting the two non-zero exponents. It follows that







70



2 O aO
2 6 6 6

2 2! 1 1! 2 2 / ' "
'o o 1 2 3 4


g(x, ... ,xm; 0)dx ...dx 1 m 1 mn


2 m H (0) 4 m-4
p (4) 0 /1
2 4 L. pJ \


2 mm
P2 /1\m (4)
2 2
TT

Although one might hope for a general expression for S such k'
hopes diminish after evaluating the next two terms in the series. With k= 3, we must consider assigning either one "three" or three ones" to the N k.'s. (The choice of one "two" and one "one" results
3

in an even powered derivative.) The assignment of the three "ones" can result in two types of products with only odd exponents on the 6's.

The first type, 63 6 can be formed by [2(m-2)](m-3)~
T h e f i s t t y e 5 i6 2 6i 3 6 i 4 c a n 3 !frm d b


different assignments, and the second type, 5616i26i36i46i56i6, can be


formed by (2) /m-2\ m-4 Idifferent assignments of the three "ones".
2 2 XL 2 /3!. Consequently,

3 ( 63 3
3S3 ( 2/ F- g(x ,x;O)dx1..-.dxm
o o 1 2

3!Dm 1 e 6 6 6 6
+ 1!111 k(2) [2(m-2)](m-3) 33...J
0 0 1 2 3 4

g(x ,...,x m;0)dx ...dx 1 m 1 m






71





0.\-l'-\~ 1 T2 3 X4 5 6S

*g(xl.. x M;O)dx 1**dx


p ~n(2) HO2 (1m
2 L r


1- H 2(0) -_H(0_ 3 m+ m(4) L- r- "O(O)"3(*mj



+m(6) H 0 l



-3 m r-m( 4m (4 371 \2, LT 2 T

Finally, for k=4, values of the k.'s equal to "three" and one" or "two", and two "ones' can both be assigned to form a product

33
result in either a 8 5 product or a j&...
1l6'2636141'516 616'2.. 18
product. It follows that


= ()[2(m-2)](m-3)j[- 2(O12 ,/2(lm


r4~~~ /m 1 1(_20-'2, H (0) -2 /m-4 + L211 2[2(m2)l(m-3)-:L JL K)


____ 414\ H~ H2 () H 0(0 5
+ (m [2(m-2)]( ) ~L -LOj

M-6
2~)







72






L Ll17 1 ~2 2 A2A) 4iL Tj ( )



4 (1~ -~4Omi(4) 4m (6) m (8)

~L 1* TT 3T


It is unlikely, from the expressions given for S 3and S 4fthat a simple general expression for the kth term exists. Although additional terms in the series could be determined, it was seen from numerical examples that the series converges quite slowly, especially for large values of rho. Some of these results are given in Table 7.




TABLE 7

ERROR INVOVED IN COMPUTING P (p) WHEN THE SERIES IN RHO IS TRUNCATED AFTER FIVE TERMS m= 5 M-_l0

Absolute S 4 Absolute S 4
Error Error


-1/10 .00001 .00006 1/15 .00006 .00002

-1/5 .00008 .00101 -1/10 .00029 .00008

1/5 .00048 .00101 1/5 .00318 .00127

1/10 .00003 .00006 1/10 .00022 .00008





It should be noted that this method of computing P m(p) is valid for negative values of rho. In fact, if the results in Table 7 are any indication of the general behavior of the series, we would expect the fastest convergence for p <0.







73



5.3 A Series Resulting from an Inverse
Taylor Series Expansion of g(u)


For positive rho, consider the expansion for Pr:m(p) given in Lemma 1:


Pr:m(p) = ( [1 G(Gy)]r [G(Gy)]m-r g(y)dy
r:m r .


2 -1
where = p (1-) Since P r:m(p) = P m-r:m(p), we can write
r:m m-r:m
P(p) = o:m(p),' so that

CO
P m(p) = G(y)m g(y)dy m2
After the change of variables, u = ey and h = 1/2, we have

21
1 1
(5.3.1) P (p) = f G(u) e dy


h-1
2 1 m h-1
= (2rr) h S G(u) g(u) dG(u)
o
0


By expanding the density, g(u), in an inverse Taylor's series, the integrand will contain only terms involving G(u). Expanding about an arbitrary point G( ), we have



[G(u) G(f)] d
(5.3.2) g(u) = gdu)
j--O 3u=jM0



= (C)[G(u) G()]
j=0







74



The function G.(u), j = 0,1,..., is given by
3


-.(u) d g(u),
j \dG(u)

and can be written


1 d ddg(u)
S j! dG(u))/ L dG(u)J


1 d ,j-1 rdg(u) dG(u) j! dG (u)) L du du

1 fd j-1
j dG(u) U


The function


dG u) u g(u) = g(u) (j+1)! o. (u)
dG(u)/j u 3+1


is tabulated by Saw (1958) for j = 0,1,...,10.

Substituting the expansion (5.3.2) of g(u) into (5.3.1), we have

h-1i
S1 ( .)h-1
= (2) h2 E .()[G(u) G(h)]
o j=0

m
SG(u) dG(u) h-I
1 C
-- 1 ko
S(2rr) 2 h5 S k(f;p)[G(u) -G(C)]kG(u)mdG(u),
o k=0

where the constant (f;p), k = 0,1,..., can be determined from the ak(f)'s. Assuming that an interchange of integration and summation

is permitted, we have







75




h-1
cc 1
2 k m
P (p (2TT) hT k [G(u) -G( )] G(u) dG(u)
k-- 0 0


Next, declining y k (F.,m) by




Y (Zm) = I M [G(u) -G(7).l k G(u) m dG(u),
k
0

we can write h-1
2
P (p (2TT) k( ;P) Yk( 'm)
k-- 0


A recurrence relation for Yk(f'm) can be found by first writing the

integrand as


k 1 k-1
G(u)M[G(u) -G(f)] G(u)' [G(u) -G(f)]

m k-1
G(f) G(u) [G(u) -G(f)]


and then integrating by parts. We have



Yk (fm) I G (u)'n+ 1 [G(u) G(f ) ]k-I dG(u)
0


G( ) G(u)m [G(u) -G(f)] k-1 dG(u)
0

1 G(u)' 1 [G(u) G( ) I kj

0


m+1 G(u)m [G(u) -G(f) ]k dG(u)
k
0

G(f) Yk-l(f'm)


k m+1 (I G(f)] -Yk
Yk (f'm) -1(f'm)






76




It follows that


(5.3.3) ,k((,m) 1 [1 -G()]k kG(() .k_(Cm).
k m+k+1 L kSince
1
yo({,m) = G(u)m dG(u)
o

1
=m+1 '


the Y k(,m), k = 1,2 ..., can be obtained easily from (5.3.3) for given values of m and G(f).

The selection of the point G(f) should be made so that the


Yk(t,m)'s are small. Since (m+l) G(u)m represents the density on G(u), we can write


1 k Y (f,m) = 1 E[G(u) -G(f)]k
k m+1

Therefore, by letting


G(f) = E[G(u)]
1
p ~ m+ 1
= (m+l) G(u)m dG(u)
O

m+1
m+2 '

y1(f,m) equals zero and y2(f,m) is minimized. However, numerical work has shown that, although yk(f,m) becomes quite small as k increases, the values of .(), j = 0,1,..., corresponding to the
J
m+1
point f that satisfies G(f) m+ become exceedingly large.
m+2
(For example, with m= 8, T= 1.28155157 and a (e ) = -3,825,025.96.)








77



Therefore, it would be better to choose the point G( ) that minimizes aj(f), j = 0,1,..., especially since the yk( ,m)'s are bounded by
-1
(r+l) Since a () can be represented by
3

a' ( ) = g( )



a'. ( ') =- I J1 j-2)


a j,i j 2,
j,(f)j-I i=0 where the a. are constants, satisfying


a 0, i + j odd,


a < 0, i+ j even,


the choice = 0 clearly minimizes 1011()I. Denoting a'j(0) by a'j, we have

32

(2T-r) 2



so that a vanishes for odd values of j. Since 0k(0;p) say k(p) is a product of the a'.'s such that Eji = k, it follows that k (p) vanishes for odd values of k, and that the series P (p) contains only
m

even terms.

In order to determine the values for the a.'s, we first need
3
i
the coefficients of u in the function a'.(u). Denoting this coeffi3
cient by a'j.i and using the recurrence relation given by Saw (1958), we have







78



=(2TT) 2
o0,0

2n
j.- j(j-1) [j-2)(j-3)a. 2
3, JJ-2,i-2 + (2ij 5i + j-3) j_2,i j3-2 ,i

+ (i+1)(i+2)a j-2,i-2], i j-2, j = 2,4,..., where c ji equals zero if either i > j-1 or i < 0. Using this relation, the values of aj, j = 0,2,...,22, were calculated. They appear in Table 8.


TABLE 8

VALUES OF a., j = 0,2,...,22
3

3 J J0'


0 .3989422803 8 -1.958451122 16 -105.6166131

2 -1.253314138 10 -4.703578753 18 -326.3330223

4 -.6562337483 12 -12.48581643 20 -1037.319292

6 -.9620889240 14 -35.44811307 22 -3373.253924



1
Substituting G(f) = G(0) = into the recurrence relation
2
(5.3.3) for yk(,m) and denoting k(0,m) by yk(m), we have
Yk(i) 1 [(I.k kk

1 fl\k k
Yk(m) Yk* y(m)]
m+k+1 \2/ 2 yk-1(m Since we need yk(m) only for even values of k, we can eliminate yk-1(m) in the above relation. It follows that







79


(i n-n) -in ~ll k (k-1) Tk (mn) 7 k=2,,
yk~m (m+k+l) (m~k) 21
km)=+ k~- k2m], k = 2,4,. .



This relation can be easily programmed to evaluate N'k (m), k = 2,4,..., for a given value of m.

Numerical examples were used to investigate how quickly the series,

h-i

(5.3.4) P (p) = (2rr) hT E 0k(P) Yk(m),
k=0


converges. As examples, we include the numerical computations for m= 10, p =1/3, 1/4. The values of Yk(m), k (1/3), and 5k (1/4) for k = 0,2,..., 22 are given in Table 9. In Table 10 is listed

h-i
2 n
(2 ) h E k ( ) k(m) k= 0


for n = 0,2 ... ,22 and p = 1/3, 1/4. The exact value of P (P) is given after n= o. As can be seen, absolute errors for p = 1/3 and

p = 1/4 are .00258 and .00122, respectively. In addition to the slow convergence of the series P (p), this method of evaluating P (p) has two other disadvantages. First, the expression for P (p) given in
m

(5.3.4) is only valid for p = h+1 h = 1,2,.... Also, the constants k (p) are difficult to evaluate for small values of rho.







80











TABLE 9

VALUES OF 'yk(m)' 8k( 1/3), 1/4),

k = 0,2,...,22, m = 10


k Yk (10) $ k (1/3) k (1/4)



0 .0909090909 .3989422803 .159154 9430

2 .0163170163 -1.253314138 -1.000000000

4 .0032092907 -.6562337483 1.047197553

6 .0006629400 -.9620889240 .8772981706

8 .0001413557 -1.958451122 1.279624120

10 .0000308241 -4.703578753 2.418906535

12 .0000068352 -12.48581643 5.323901882

14 .0000015356 -35.44811307 12.95550088

16 .0000003486 -105.6166131 33.85865868

18 .0000000798 -326.3330223 93.33839437

20 .0000000184 -1037.319292 268.1907545

22 .0000000043 -3373.253924 796.5361116






81






TABLE 10

THE FIRST n TERMS IN THE SERIES P (p),
m
m= 10, p =1/3, 1/4, n 0, 2,...,22


n p = 1/3t p = 1/4



0 .128564 .157459

2 .056070 -.020116

4 .048604 .016459

6 .046344 .02278S

8 .045362 .024757

10 .044848 .025568

12 .044546 .025964

14 .044353 .026181

16 .044222 .026309

18 .044130 .026390

20 .044062 .026444

22 .044011 .026481

co .043753 .026603

1 1 n

tTabulated entries are (2rr)T 2r Z k (1/3)yk(10).
k0



Tabulated entries are (2r) 3 Z S k(l/4)v k(10).
k-=0







82



5.4 Using Moments of Extreme Order Statistics


If we let f(u;m) represent the density on the largest order statistic in a sample of size m from a normal distribution, then


f(u;m) = G(u) du

mn-i
= mG(u)m- g(u), -0

Using the representation of Pm(p) given in (5.3.1), we can write h-1
(2rn) h 2 h-1
P (P) + g(u) f(u;m+l) du.



The integral, say I, above can be simplified by successively integrating by parts. For example, after integrating by parts twice, we have


I = Sg(u)h-1 dG(u)m+1
-CO


= g(u)h-1 G(u)m+1 (h-1)u g(u)h-1 G(u)m+1 du



= o + u g~Cuo2d~u1
h-1 COguh-2 dGum+2 = 0 + IIu g(u)h- dG(u) m-C2


(h-i) (h-2) c 2 h-2 m+
(h-1)(h-2) (u-2_1) g(u)h-2 G(u)2 du m+2


(h-1)(h-2) 2_ h-3 3
S(m+2)(m+3) (u2 -1) g(u)3 dG(u)m+



It can be shown by induction that after integrating by parts k times,







83




= (h-1)(h-2)...(h-k) h-k-1
(m+2)(m+3)... (m+k+1) -~ k g(u) f(u;mk)du



where Hk(u) is the kth order Hermite polynomial in u. Therefore,

after h-1 integrations, we have


h-1
2 C
(5.4.1) P (p) = (2) h(h-l)(h-2) ...21 f(u;m+h)du
m (m )(m-+2)... (m+h) fCm -1u

h-1
2
(217) E
=E [R_ (U) ],
I (m,,h )h-1 U'
-f h h U(m+h)



where Um+h) is the largest order statistic in a normal sample of
(r-ih)

size m+-h.

For example, with p = 1/3 and p = 1/4, expression (5.4.1)

simplifies to


2n
P (1/3) = E U
m (m+1)(m+2)
U(2
U (m+2)


and
4 r3 T 2 1]
P (1/4) = E U -_12
m (m+l)(m+2)(m+3) U (
U(m+3)


respectively. Using the table of moments of extreme order statistics

computed by Ruben (1954), expression (5.4.1) can be used to determine

P m(p) for values of m and p satisfying p > 1/12 and m + p -1<51,

where -1 is a positive inter.
where p is a positive integer.









































APPENDIXES







APPENDIX 1
i k (p)
p
k 1/ 2 1/ 4 1/ 5

0 1.OCCOC 1.COCCG 1.CCCOC 1.000co
2 C., CCGC -C.17548 -0.25871 -C.30772
4 O.lljocl j: -C-CCE77 0.'2771 G.C6523
6 C.CCCCC -C.Colep C.CC243 -O.CO299
3 C. )Ccoc -C.CCC52 C.CCC45 -C.COC41
io O.r cocc -C.CCC21 C.rCC12 -0.000ce
12 C.I)OCCC -C.rlcl.,lc .*CCCQ4 -O.CO002
14 C.GCCCC -C.Cl"CC6 C.CCCG2 -C.Cccci
16 C.OCCCC -C.CCCC3 C.Cccol -C.COOGO
C.Ococc -C-CCCC2 C.CCCCC -C.00000
2C 0.lcllcr -C.OOCC2 O.CCCGC -C.COOOO
22 ().OCrCC -C.CCCC1 C.CGCCC -C.Cocco




1/

0 1.0ccc.c i.cccrIrl 1.GCCCC i.COCCO
2 -0.3ziClC -C.36311 -C.38C32 -C.39368
4 C.IS766 C.12483 C.14757 C.16b73
6 -C.01365 -C-C26G8 -C.C4C7C -0.05415
8 O.OCC02 C.OG21il C-CC647 C.C1172
1L- C.OCCC4 C.CCCC6 -C:CCC32 -C.CO134
12 C. Ccul C.CCCCC -C.CCC02 C.CCO02
14 O.OCCCC -C.CCCCC -C.CCCCC 0.00000
16 c.ocl;oc -C-COCCG -O.CCCCC C.00000
18 0.0ccc: -C.CoCrC
%. -C.CCCCC C.Cocco
20 C.OCCGC -C.CCCCO -C.Ccccc C.Goooo
22 O.OCCCC -C-OCCCC -G.CCCCC O.Coooo




1/10 1/11 1/12 1/13

0 1.,Ncoc I.CCCCC 1.cccck,4 1.COOCC
2 -C.4043 -C.413C7 -C.42C33 -0.42647
4 O.lb3G 0.197C4 C.2CSIS C.21981
6 -C.066G6 -C.07ES5 -().Cgccs -C.10040
8 C.01776 C.Ul1255- C.C3C96 C.C3771
10 -C-OC524 -C.CC796 -C.01107
12 C-CCC61; C-CC140 0.00239
14 C.CCCCC -C-CCCC3 -C.CCC14 -C-OOG34
16 -C.CCCCC C.CCCOO C-COCC2
16 -L..CCCO C.Cccoo 0.00000
2 : -0. Oc tic) c c .. c I, c C. 0 0 c *C c 0 c -O.Cocco
22 -C.CCCCC C.CCCCC -O.COCCC


85






86
APPP.,TDIX 1 (Continued)


p

k 1/14 1/15 1/16 1/17

0 1.occ% c I.CCCCC I.CCCCC
n
IL .43172 -C-43E29 -0.44C28 -C.44350
4 0.22917 C.23749 C.24491 C.25159
6 -6.ic;;2 -C.IlE72 -0.12686 --C.13439
ji C.3444C C.05CS6 C.C5733 t,'.G6348
10 -C.01447 -C-CIEC9 -C.C2185 -0.02570
12 0.00363 C-00512 O.rO682 C.00870
14 -C.CUC6 -cojolic -C.CC168 -C.C0239
16 0 3 C 0 7 C C 0 C 17 C'. w C C 3 1 C M )51
is -C*OOCOC -C-CCCC1 -C.CCCC4 -C.COI--08
2r, 0 -,: G C 0 C-COCCO c "I ccoo C-CoCol
22 -C.--CccC C.CGCCC -G.r-cccc -O.COCCG




1/18 1/is 112 C 1121

0 I-Pcoo 1.oCrCC
2 -0.44692 -C.44S72 -C.45223 -C*45451
4 0.25761 'C.263CS C.26PO7 0.27263
6 -0.1413E -C-14727 -C.1539.: -C.15953
O.-Ot941 C.07510 C.108C55 O-OS577
-C.,O 2 q6'L -C.G3352 #I C 3 7 4 2 -C*04129
12 0 0 1C 7 3 C.0128S C.C15,16 C.01751
14 -C.:C323 C C Cl/ 19 C. C C 52 6 -^-.00644
16 0 -,0 C 7 E C-O;112 C-CC154 C-002C2
13 -C.rlllr,15 -C.CCC24 -C-CCC37 -C.CO053
2 1 O.CCOU2 C.COCC4 CI.CCCC7 C-co ll
22 -0.0c0c". -L.C3,^CC -C. cccol -C.CCC402




1/22 1/23 1124 1/25

C 1.CCCCC 1.CCCCC I.CCCOC 1.00, Oo
2 -0.4565F -C.45E47 -C.46C2C -0.46179
4 0.27682 C-28C68 C.28426 C-28757
6 -0.1647S -C.161;71 -C.17433 -4.17866
3 O.n,9076 C.O';553 C lcccs 0. 10445
C) -C .0 45 i c -C.C4EE5 -C,. C5253 -0.05613
in
C.'1991 C.C2237 C-C2485 C-02735
14 -C. '-C771 -C.C^SC6 ".ClC48 -C.01196
i 6 ^j.O-'-25F C.CC22C OT. r- C 3 9 C O.CO465
is -0.^CC,72 -C -COCS8 -C-CC126 -0-GO159
2 .? 0.^CO!7 C' 0 C r 2 5 C-',. C C C 3 5 C.M 47
22 0 C 0 C, 0 3 C 0 C C C 5 -C.CcCC8 -CoCCO12







87
APPENDIX 1 (Continued)


p
k -/3-1/ 14 .-l/ 5 -11 6

i.co i.occoe 1.ccccc i.ooocc
2 -0.62452 -C.7412G -C.6q228 -C.b5990
'4 2.44596 1.51C71 1.15617 C.G6895
6-4.C4666 -2.53547




-1/ 7 -1/ 8-1/ C; -1/ic

31.C)CCoC 1.CCCCC 1.CCCojC 1.Clocco
2 -0.6368S -C.61S68 -0-60632 -C.59565
4 0.353316 C.775C5 C.71e5t r% 67592
6 -1.86571 -1.4S315 -1.2586C -1.09863
?7.0626C A.466S8 3.2C847 2.48636
iG-12.72389 -8.09368




-1/11 -1/12 -1/13 -1/14

0 I.C0ccrl cco 1.CCC CCCGc 1.OCO
2-n..58693 -C.57S67 -0.57353 -0 .56827
4 C.64263 C.615S2 0.594C3 C.57576
6 -0.98317 -0.89624 -%O.E2862 -C.77462
6 c,.J72C 1.713E4 1.46E57 1.32C03
ic -5.7169C -A.32794 -3.43837 -2.83220
12 23.39377 1A.93534 1C-.42892 7.75,338
14 -43,61756 -27.917CC




-116-1/17 -1/18

0 .-o1 OCC CC 1 VCCOO I .COcco
2 -0.56371 -C*555972 -0.5962C -C.55308
4 0.5602S C.547lC3 0.53553 C.52546
6 -0.731~5F -0.694CC -C.66318 -0.636b7
81.1898E 1.08676 1.rC33C 0.93454
10 -2.39876 -2.076E4 -1.83032 -1.63672
i2 6.03311A 4.86226 4.C31C7 3.41832
14 -19-31964 -14.15336 -IC.82316 -8.57463
16 62.1604E 5Z.68241 36.2039,% 26.21621
18 -155.S7371 -I100.15185







88
APPENDIX 1 (Continued)


p
k -1/19 -1/2C -1121 -1122

c1. 0C.CC C 1.cccco 1 .,CCOC 1.00000
2 -0.5502E -C*54j77 -C0.54r54q -0.54342
0 O.5165F C.50868 0.50162 C.49527
6 -C.61415 -C.59436 -C.57696 -0.56155
6 0.87702 C.82827 0.78648 0.7503C

12 2.953C,3 2.5GC8G 2.30307 2.C7017
14 -6.97997 -5.81267 -4.933-'43 -4.255C0
16 19.77901 15.42370 12.3584C0 10.12928
18 -6S.43843 -4S.0852,1 -36.4-S633 -28.16580
20 297.927CS 1S1.51C78 130.2634C 921.67789
22 -571.91135 -367.95599




-1/23 -/4-1/25 -1/26

0 1.0cccc 1.C~lCCC 1.cccOc 1.COZCO
2 -0.5A153 -C.53SEC -C.53821 -C.53674
4 0.48951 C.48429 0.47(;52 v-.47514
6 -C-54t762 C.53550 -C.5243S -C.51432
a 0.7187C C.69CE9 0.t6622 "0.64422
I -1.0E596 -1.C2140 -C.36'543 -0.91650
121.87876 1.71S31 1.53483 1.47034
14 -3.72C57 -3.292CO -2. 1421;1 -2.65465
16 9.462,;2 7.1879,7 6.119143 5.39946
18 -22.2593C -17.98S90 -14.82C64 -12.41346
20 68.40 097 52.04764 40.63786 32.43603
22 -249.31169 -176.16453 -128.EE4259 -97.09741
24 1102.3E011 70S.76567 479.32931 336.70025
26 -2132.22218 -1373.66691









APPENDIX 2
d k (M)

m
k 2 3 5

0 C-323333-1-2 C.25rrCCCC 0.2cccccoc C.16666667
2 ').16666667 0.25CICOCCC 0.28571429 0.29761,3C 5
4 0.28571423 0.29761905




6 7 9

c 0.1428571/1 G.125CCCCC 0.111ILill C.lcocccoo
0.2976190 C.2SI66667 C.28292E28 0.27272727
C.0584415 C.,071;51#545 0.09730210 C.11328671
6 0 C C' I r 8 2 2 5 ODC3787E8 0. COE '-t8C8 1 C.01363636
O.CC808C81 C.01363636




10 11 12 13

0.09314c9cs 0*08333333 0.076;2308 C.07142857 2 0.26223776 C.2518315C 0.24175824 0.23214286
4 Ct.'?587413 C.la5GasCl 0.143qEE36 G.15021CG8
6 v.62C,45348 C.52696078 C.C34^15573 0.041110-13
a Cl.Cl3q2cl2 C.OC185560 O.CC318103 C.CC488722
V, r.CCO' C541 O.CC:32977 O.CCC:931; 0.00021874
0 c c c IC 9 3 19 C.OC021874




14 15 16 17

c C.36666667 0.0625CCCO O.C5582353 C.05555556
7 7i,.223D3922 0.21A46C78 O.2C639E35 C.19863041
4 0.15495356 C-15847 --23 0.16iC,99C71 C.16267943
6 C.)4'V;6182 0*054502C6 O-C6C66317 C.0664C778
8 O.rl'r6q4l27 C.OC'329634 C I 18 S S31 0.01469616
ll O.CCG4281-1 C.OCC742C2 O.CC117258 C.00172896
12 0 C C' C C 17 2 3-OCCO2337 0 .,-- C' C; t' 5 15 7 G-0C"-)09 437
14 CI.CCOOC002 O.CCCCGC19 O.CCC- 'CC77 C-OCOC0232
16 O.CCCCCr77 C-OC000232






89







90

APPENDIX 2 (Continued)


m
k 18 19 2C 21

0 i .C;52b315E 0.05ccocce 0.04761SO5 C.C4545455
2 O.IS172932 C.lE5C6 G4 0.17HOC670 0.17292490
4 Q.i6358986 C.16114 55 0.16414455 0.16377318
6 C.07172C41 Q'.0766OC79 O.'O81C5904 f .08511199
C,.C176353G 0.02067C,05 O.C2375868 C. 2686559 111- O.C02eiJ45E 0.0%'322t3l; C.C0416566 C.00521391 12 O.CCCI.73CS G.CCC27E95 O.CCC42265 C.Or. )60912
14 v. CCOO C 570 C CC CC l 2 11 C.00C023CT C.OCC,4038
16 C-CCJ'jCCv7 O.PVCCCOC22 G.OCC CC59 C.OCOGC137
1 O.CCO rror O.CCCCCLCC O.CICCCCO1 C.0toOCOL02
2,3 O.CccccrIol C.O,--CCO ,02




22 23 24 25

c .CA34782 C.04166E67 O.C4CCCCOC U'.03846154
2 0.1673913C 0.16217GA9 0.157264S6 C.1526251.
4 -n.16309922 0.16217,349 0.16106101 0.15978275
6 1,08878061 C.Oli208EI2 O.'l'G5C5871 C.r)977166C
8 3.02996"'17C 0.0330IS13 O.C3602C93 C.03834997
1 -1
%., 3.C0637367 (;.CC763419 O.C'08G7407 C.01038666 12 C .(-'iliS423Z C.OC112522 O.CO145975 C.OC164685
14 ;.cCVi6604 G.3CC10212 O.CCC15%82 C.CC021423
16 O.CCOOC284 O.OCCO0536 O.CrCCCS42 C.O''CO1557
C.CCCCQC15 O.CCC','Cr,33 C.CCOC0066 2c C, C t,0,, r
O.OCCCOLCO G.ccccccol C-OCCCGO01 22 O.CLCC*CCCO O.Cclljocc)c C.Occccooc
24 O.Cccccccn C.ccccocco









APPENDIX 3
P m (p


m 1/ 2 1 / 3 1/ 4 1/ 5

2 0.331-33 C.304C9 C.29C22 C.282C5
3 0.25CCC G.2CE13 O.le532 0.17307
4 0.2CCOC C.14S74 C.12648 C.11301
5 0.1666? C.11i,13 C.CGC66 G.G7741
6 (,.14286 C.09C12 O.C6748 C.05508
7 0.125 "'C C.07311 O.C5176 CO.04043
3 0.11111 C.06C61 O.C4C67 C.C3044
061CCOC C.C5113 C.C3262 .C2343
C.09C91 C.04375 0.02660 0.01836
C.0833 ;.C37SC 0.02-202 C.01463
12 C.0761 2 C.C3318 C.ClE45 0.01182
13 0.071,43 C.02 3C G. ;1564 O.GO967
j/. 0.06667 C.026C8 0.01336 C-00799
i5 0.0625C C.02-223E C.,;1155 6.OOt68
16 C,35882 C.C21C8 C.ClC04 O-CO563
17 0.,.-)5556 C-ClS12 C.CCE79 C.CC478
18 C.01742 C.CC775 0.00409
19 C.01594 COC0687 C.C0352
2C C.04762 C.01465 C.CCtl2 C.C0305
21 mo)4545 C.01351 C.CC548 C-00266
22 C. 34349 CoC1251 O.CO492 O-OU233




1/ 6 1/ 7 1/ a 1/ 9

2 f .2766 C.272el 0"26S95 0.26772
3 ().164G8 0.15S22 0.15492 OoI5158
4 0.10422 0.09EC4 C*C9345 C.0899C
5 CoO6R92 C.062C6 C.C5E75 C.05546
6 C.04733 C.042C5 C.C31E25 0.03538
7 r .,-, 3 35 2 C.028 2 C.%C2566 0-02323
3 Go 2437 C.C2C42 O.C1766 C.01565
9 .01812 C.01414 C.C1244 O.CIOV79
).D1374 C.CiCE6 O-Cf.894 C-C0758
C.CIC5 CoGO814 C.CC654 O.CC543
12 C- C82S CoOC62C C.dC486 C-C0395
13 C.OC65E C.CC479 C-C0366 CoC0291
14 1%1 054E C-CC274 C-CC28C C-C0218
15 C. .DC420 C.00296 C-00216 CoOO165
16 O.')C351 C-00237 C.CO169 C-00i26
17 O-OC29C C.CGIS1 C-CO133 r1;.oojs3
13 C-00242 C o 00 15 5 C.CC1C6 CoCC076
19 C.0C2,-'3 C-OC127 C-CCO85 C-COC60
2 '13 1 o: 0 17 2 c.Colrq, C 0 c c c 6 G C.COC48
21 0. jG146 C.Cccea C.CCC56 C-COG38
22 O-JC125 C-COC73 C-CCC46 O-CO031
91




Full Text
53
(4.3.3) u = 2 K (p ,0,k)
j=0 3 (n+2)
= E H.(p ,k) n(j) ,
J V
where o = and, for convenience, we have replaced H.(p,,,0,k) and
* v rn-1 J v
r by H.(p ,k) and n(j), respectively. The constants, H (p^,k),
(nf2)J J V J
are tabulated for k = l,2,3,4,j < 5, and py = .50, .55 95 by
Flora (1965). It follows, since the moments are functions of
that
6
k V
E C.(pv>k) n(j),
j=0 3
where the constants C.(p^,k) are functions of H.(p^,
J *3
In order to find the C.(pv ,k), we must first
J
§4i as functions of the Letting <^(t) represent
(v)-Y, we have
kJV
istic function of Y
k).
express the
the character-
Since
Cp(t) = E
Y and Y Y
(pit) = E
t(Y
are independent
Lt (Y
e
we can write
e(/7]


TO MY PARENTS


CHAPTER 4
APPLICATION: A TEST FOR NORMALITY
4.1 Introduction
Consider using a random sample, Y ,Y ,. .,Y from a continuous
12 n
distribution F to test the hypotheses:
H : F is a normal distribution
o
H : F is a skewed distribution,
a
As David (1962) suggested, one might consider the number of observa
tions larger than the sample mean as a test statistic. Letting
if Y. > Y
J
if Y. < Y ,
J
for j = l,2,...,n, the test statistic, Y, can be expressed as
n
(4.1. 1) Y = Z cp. .
j=l J
Without loss of generality, we can assume that the variables in the
sample have been standardized to have zero mean and unit variance.
Then, using the representation of Y given by (4.1.1) and assuming that
H is true, we can find the mean and variance of Y. For the mean,
o
we have
44


This dissertation was submitted to the Department of Statistics
in the College of Arts and Sciences and to the Graduate Council,
and was accepted as partial fulfillment of the requirements for the
degree of Doctor of Philosophy
March, 1972
Dean, Graduate School


7
Var
cov
u+eV
x:
i
1+ 9
= 1,
x'
2 Var (V
(1+62)2 (1+02)2
i cov (X! ,x')
i J
1+0
1+0
= p
Hence, we have
P to)
r: m
P(X >0 X >0; X <0,...,X <0)
1 r r+1 m
0
1 > 0 > o-
2 f 2 ?
(1+0 )2 (1+0 )
Xr+1 <0 -J <0
(l+02)^
(i+e2)^
-C)
P(x'>0 x'>0; X' <0 x'<0)
1 r r+1 m
= () P(Y -0Y >0,. . ,Y -0Y >0;
vr / 1 o r o
Y -8Y r+1 o mo
This last expression can be simplified by noting that the variables
Y ,Y ,...,Y are mutually independent with common density and distribu-
o 1 m
tion functions
g(y)
G(y)
_ CO < y < 00 t


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii;L
LIST OF TABLES vi
LIST OF FIGURES vii
ABSTRACT viii
CHAPTER
1 INTRODUCTION 1
1.1 Introduction 1
1.2 Definition of P (p) 2
r:m
1.3 A Transformation Simplifying P (p) S
r: m
1.4 Summary of the Results of This Dissertation ... 12
2 AN EXPRESSION FOR P (p ) INVOLVING TCHEBYCHEFF-
r :m
HERMITE AND LEGENDRE POLYNOMIALS 14
2.1 Definitions and Properties 14
2.2 The Fundamental Result 18
2.3 The Integral J (p) 20
3 NUMERICAL RESULTS 26
3.1 Exact Results 26
3.2 Evaluating the Integral J (p) 31
xC
3.3 Computing P^(p) 38
3.4 Accuracy of the Results 42
4 APPLICATION: A TEST FOR NORMALITY 44
4.1 Introduction 44
4.2 The Null Distribution 47
4.3 Approximations to the Null Distribution 49
iv


10
Since p >
-1
TO-1
2 -mp
rrtp =
m
m-1+1
= 1
and the integral
m
e e g(y)dy
J
converges. Therefore, interchanging the order of integration is
permitted and we have
R
-C)
f*
P
J '
[r,m] L-co
m
00 -y(cp .Z t ) 9 9
J=1 J 1 i(l-irtp2)y2
v/2tt
dy
g(t )...g(t )dt ...dt .
1 m 1 m
The integral in brackets multiplied by (1-nip ) is the characteristic
function of a normal random variable with zero mean and variance
, 2n -1
(1-mp ) .' Hence,
m
m
-C)
m
J (2tt) 2(l-ntp2)"^ e
Lr,m]
j=l J
(cp E t.)
j=l J
~T
e 2(1-^ > dt,...dt
1 m


17
2 i
[l-2s(2p-l)+s ] 2
2 --i
[l-4sp+2s+s ] 2
2 i
= [(1+s) 4sp] 2
- (i)-1 [i -¡2-J
-1
(1+s)
,-l (-\ (4sp)
= (1+s) E (-1) 1 f j 5-t
i=0 Kl (lts)2i
/ _i\ (2Z) l
But ( 4 ) may be written as (-1) ^-r
v £ a:)4
since
(-§X-§)(-*£)
(
i j
, ... ( 2£-l)
= (-D 1
= (-1)
2 x:
£ (2£) !
I
2 £! 2.4 . 2£
= (-D
X (2X):
4(X:)2
Substituting the above result into the expression for [l-2s(2p-l)+s ]
we have
2i *
1
[l-2s(2p-l)+s2] = E
(2X): x i
(sp)
-to (X!)'
(ln-S)
2in
00
- (2X)
Z_i
x=o (X
(sp)£ S (-l)J (2Ui) SJ
:)2 j=o v '
^ r (2X+J)! _X _j + X
E E, (-1) ¡~r~ ft P s .
X=0 j=0 J-l-l-
which completes the proof.


APPENDIX 2
d (m)
k
k
2
3
m
A
5
C
0.22333333
C.25C0CCCC
O.2CCCCC0C
0 1666667
2
0.16666667
0.25CC0CCC
0.2 35 7142*3
0.297619C5
A
0.28571429
0.29761905
6
7
8
9
C
0.14235714
C. 125CCCCC
0.11111111
c.icoeocGo
4.
0.2976 1905
C.25166667
C.28232628
C.27272727
4
C.05844156
0.07554545
0.09790210
0.11328671
6
0.CC108225
O.OC378788
0.00808081
C.01363636
8
0.CC808C81
C.01363636
10
11
12
13
0
0.C909C9C5
0.08333233
0.07652308
C.07142857
2
0.26223776
G.2518315C
0.24175824
0.23214286
4
0.10587413
0.13598901
0.14358636
G.15C21CG8
6
u.02005348
C.02696C78
C.C3405573
C.04111013
8
0 .COG 92013
C.0C1S556C
0.C0318103
C.00488722
10
0.CC00C541
0.CCC02577
0.CCCG5315
C.00021874
12
0.CCCC5315
C.0CC21874
14
15
16
17
/*>
0.06666667
0.0625CCCC
0.C5882353
C.05555556
2
C.22303922
0.2 1446C78
0.2C639835
C.19863041
4
0.15495356
0.15847523
016099C71
C.16267943
6
0.04796182
0.054502C6
0.C6C66317
C.0664C778
8
0.0^694127
C .00529634
C.C1185531
C.C1469616
10
0.CC042372
C.0CC742C2
0.CC1 17258
C.00172896
12
0.CC0CC872
0.0CC02337
O.CCOC5157
0.00009937
14
0 C COOCOO 2
0.CCCCCC19
0.C0CCCC77
C.GC000232
16
0.CCCCCC7 7
C.CC000232
89


15
(2,1.5)
L (t)
c
Lx(t)
L, (t)
k
= 1
t
2k-l
k ^ 2.
The following result due to Saw and Chow (1966) connects the
terms of a binomial series with the Tchebycheff-Hermite and Legendre
polynomials. For any p,
m
(2.1.6)
I
r=0
m\ r,-, sm_r / X
TJ P (1-P) ck(r,m)
ml (k!) t /0_
(m-k).'(2k): Lk(2p 1}
The importance of this result to the next section justifies the inclu
sion of the following proof.
After substituting c^ir.m), as defined by (2.1.1), into
equation (2.1.6) and multiplying the equation by (m k) -/-k) the
(k!)
equation to be verified simplifies to
m k
£ <-J 0 () pr<1-p>m" = i:i.k(2p-i)
Notice that (j^^j) eciuals zero unless r 2: k j. Thus letting Q
represent the left-hand side of the above equation and changing the
order of summation, we have
m
Q I (.1)j [X (>'a-P)"-r]
j=0
r=k-j
Letting r = r-k+j, the sum in brackets simplifies to


13
An application of the results of the first three chapters is
given in Chapter 4. The statistic suggested by David is used for test
ing the hypothesis that a population is normally distributed. The null
distribution is discussed and numerical resuJts are given for sample
sizes not exceeding 22. Approximations to the null distribution are
also given.
Finally in Chapter 5 we discuss three alternative methods for
computing P^(p). However, none of these methods can be used to obtain
numerical results as readily as the method discussed in Chapters 2 and 3.


n
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
TABLE 1
STANDARD DEVIATION OF Y
n
.50000
27
1.56581
.59242
28
1.59456
.66760
29
1.62281
73389
30
1.65058
.79416
31
1. 67788
.84994
32
1.70475
.90214
33
1.73119
.95140
34
1. 75724
.99818
35
1.78291
1.04283
36
1.80822
1.08562
37
1.83317
1.12678
38
1.85779
1.16646
39
1.88208
1.20484
40
1.90607
1.24202
41
1.92976
1.27811
42
1.95316
1.31320
43
1.97628
1.34738
44
1.99914
1.38071
45
2.02173
1.41325
46
2.04408
1.44505
47
2.06619
1.47617
48
2.08806
1.50664
49
2.19701
1.53651
50
2.13112


49
n
Var (U.) = - Var (Y. Y)
J n-1 j
and
-Ar (1 -)
n-1 n
= 1
n
corr (U.,U ) = - cov (Y. Y, Y, Y)
J k n-1 j k
= Ar (- i)
n-1 n
1
n-1
Hence, for m ra-1
UU,...,U have a multivariate normal distribution. Therefore,
1 z m
P^(p), k < m, is defined and can be computed, using the method discussed
in Chapter 3. Furthermore, since P(Y= n) = 0, we can set P (p) equal
n
to zero and use relation (3.3.3) to find P (Y= r). That is,
n
/ \ n-r /
(4.2.1) Pn(Y = r) = (r) E (-DJ (n~T ) K. ,(p) >
r+J
where o = - and P (p) =0. Using equation (4.2.1) and
n-1 n
the results of Chapter 3, the null distribution on Y was obtained.
The results, for n < 22, are given in Appendix 4.
4.3 Approximations to the Null Distribution
(H)
David (1962) has shown that the asymptotic distribution of
is normal with mean zero and variance Hence, for
4 2tt
large n, we should be able to approximate the distribution on Y, using
a normal distribution function. In particular, the critical values of


19
Finally, applying Lemma 1 to the left-hand side of the above equation,
we have
(2.2.3)
m
Z c (r, m)
k=0
P
r: m
(p)
m! (k*.)"
(m-k)! (2k)! k^D;
An alternate expression for Pr.m(p) can be obtained by noting
that for fixed m the set of points P (p), r = 0,1,...,m, lie on
r: m
a polynomial of degree at most m. Hence, for some constants, say
e ,e_,...,e we can write
o 1 m
m
(2.2.4)
p (0)
r: m
E e c (r m) .
j=0 J J
Multiplying (2.2.4) by c (r,m) and summing over r yields
m mm
(2.2.5) E c (r,m) P (p) =
k r:m r
r=0
Using the properties (2.1.3) and
side above reduces to
E E e .c (r ,m) c, (r ,m) .
r=0 J=0 J J k
(2.1.4) of c (r,m), the right-hand
m 2
k E ck(rm)
r=0
= e.
ikM2
\2k+iy
pky
V k /
Hence, the constants e ,e ,...,e can be determined by equating the
o l m
right-hand sides of equations (2.2.3) and (2.2.5). That is,
CkM2
1 ; \2k+lJ
2k N
k )
ml (k!)'
(m-k)!(2k)! k
r J, (P)


32
Figure 1.
The
P =
i r u2~jh-1
function h2 L^[2G(u) 1} le 2 i
1/30.


CHAPTER 5
OTHER METHODS OF EXPRESSING P (p)
m r
5.1 Introduction
Before discovering the expression for P (p) involving
m
Tchebycheff-Hermite and Legendre polynomials, three other methods of
expressing P (p) were used in attempting to obtain numerical results.
Each of the first two methods, outlined in Sections 5.2 and 5.3,
expresses P (p) as an infinite series. However, in each case, not
m
only is the series slow to converge, but no workable expression can be
given for the kth term of the series. Therefore, these methods are
not useful in obtaining accurate numerical results. In Section 5.4, we
give an expression for P^(p) involving the moments of extreme order
statistics. However, this expression can be used only for limited
values of m and rho.
5.2 A Power Series in Rho
Using the definition of P (p) given in Chapter 1, we can write
00 00
P(p ) = T f g(x, ,x ; p) dx ... dx
m 1 m l m
o o
where g(x ,....x^; p) is the multivariate normal density on the equi-
correlated variables X ,...,X Since, when rho equals zero, the
1 m
density function simplifies to
62


20
After slight simplification, we have
[
' (k:) (m+k+l).
) ,
k = 0,1,...,m .
Letting b (m) denote the constant in brackets and substituting into
k
equation (2.2.4), we have
m
(2.2.6) P (p) = E c (r, m) b (m) J, (p ).
r:m r k k k
k=0
Before investigating the integral J (p ) we should comment on
K
the utility of the expression for P (p) given by (2.2.6). Most
r: m
important, by defining J^.(p) appropriately, the expression is valid
for both positive and negative values of rho. Next, the integral J (p)
K
does not depend on r or m. Hence, for a given value of p, only one
set of values, J (p), k = 0,1,..., is needed. Also, as will be shown
K
in the next section, J (p) = 0 for odd k, thus decreasing the number
K
of terms in the series by one-half. Furthermore, for large m, the
series may be truncated without serious effect, since the factor
c (r,m) b (m) approaches zero as k increases.
iv K
2.3 The Integral J (p)
K.
Consider the integral
CO
VP) = f LkCl-2G(9y)] g(y) dy
_ 03
defined for k = 0,1,... in the last section. Using the recurrence
relation (2.1.5), it can be seen that the Legendre polynomial L^it) is


27
The value of P2(p) can be found in closed form by integrating the
original expression for P (p) given in equation (1.2.8). With
m
m = r r 2, we have
CO 00
r*
?2 (P >
^Ji
(2tt) 1 (1-p 2)~ ^ e
Xl+4-^XlX2
2(l-p2)
dX!dX2
Making the transformation
VPX2
U1 2 j
(1-p )'
U2 = X2 '
it follows that
2
i ) u, + pu.
x1 = (1-p )~ ux + pu2
x2 U2
and the Jacobian of the transformation, J(x ,x -* u ,u ), is
1. Ct -L
J(X1X2 UlU2) =
(1-p2)4 P
= (1-p2)4 .
Therefore,
P2(P)
r
j
2 9
(1-p )f
u
_1_
2n
i, 2 2.
-?(U1 + U2)


99
Steck, G. P. (1962). Orthant Probabilities for the Equicorrelated
Multivariate Normal Distribution. Biometrika, 49, 433-445.
Steck, G. P. and Owen, D. B. (1962). A Note on the Equicorrelated
Multivariate Normal Distribution. Biometrika, 49, 269-271.
Stuart, A. (1958). Equally Correlated Variates and the Multinormal
Integral. J. Roy. Statist. Soc. Ser. B. 20, 373-378.
Teichroew, D. (1956). Tables of Expected Values of Order Statistics
and Products of Order Statistics for Samples of Size 20 and
Less from the Normal Distribution. Ann. Math. Statist., 27,
410-426.


16
m-k+j
l
m:
/ n (m-k+j-r')!r'I (k-j) !
r = U
r'+k-j m-k+j-r
p (1-p)
m-k+j
- Cc-j) ^ I
n-kiJ) pr (l-p)-k+J-r
r = 0
/ m \
- U ^
Therefore Q reduces to
k
Q £ (-l)J -k-J
j:(k-j):(k-j):
rrr p
j=o
Since by definition
E skL (2p-l) = l-2s(2p-l)+s |
k=0 k L J
it remains to show that
1 k r 2l
-=r E s Q = | l-2s(2p-l)+s
m k=0 L J
-i
Making the change of variable JL = k j the left-hand side of the
above equation becomes
l
JUO j=0
(2i+j): /j+j
j
TrHJT p s
2 -i
However it can be shown that [l-2s(2p~l)+s ] also reduces to the
above sum.
We have


39
A further simplification in the computation of P^ip) can be
made by combining the constants b (m) and c (m). Letting d (m) denote
K K K
the product, we have
dk(m) = t>k(m) ck (m)
m! (2k+l)! ml(k!)2
(k!)2(m+k+l)! (m-k)!(2k)!
(m!)2(2k+l)
(m+k+1)!(m-k)!
The constant d (m) can be computed for even values of k by the recur-
1C
rence relation,
d (m)
o
1
m+1
(3.3.4)
dk()
2k+l m-k+2 m-k+1
2k-3 m+k+1 m+k k-2 m
k = 2,4 m.
Finally, combining the above results, we have the following computing
formula for P (p ) :
m
m
(3.3.5) P (p) = E d, (m) J.(p).
111 k=0 k k
k even
The proofs of (3.3.2) and (3.3.3) follow. To prove (3.3.2),
we first let r = m in the definition of c (r,.nt) given by (2.2.1).
K
Then, writing combinations as factorials and caracelling like terms,
we have
- Tsyr j0 <->j c;) (2k;j) rr:
(k!)2
m:
(2k-j)!
-iEn (_1) -
J=0 tik-j): (k-j): j:
(2k):
(m-k)!


Next, we use a result connecting the terms of a binomial series
v/ith Tchebycheff-Hermite and Legendre polynomials to obtain a finite
seri.es expression for the probability. Although the general term in
the series involves an integral which cannot be evaluated in closed
form, this integral depends only on the correlation and can be eval
uated by numerical integration for both positive and negative correla
tion. The numerical results are included in the appendixes.
As an application, the number of observations larger than the
sample mean is used as a test statistic for testing the hypothesis
that a population is normally distributed. The small sample null
distribution is derived and numerical results are given. Approxima
tions to the null distribution are also discussed.
Finally, we discuss three other methods of expressing the
probability that the entire set of equicorrelated normal variables is
positive. Two of these methods express the probability as an infinite
series. However, in both cases the convergence is quite slow. The
third expression, involving moments of extreme order statistics, can
be used for obtaining numerical results only for limited positive
values of the correlation.
IX


LIST OF TABLES
Table Paee
1 Standard Deviation of Y 48
2 The Cumulative Distribution of Y, n = 19 51
3 C.(p ,2) 57
.1 v
4 C.(pv ,3) 58
5 C.(p .4) 59
J V
6 Moments of Y^^-Y,n:=19 61
7 Error Involved in Computing P (p) when the Series
m
in Rho Is Truncated after Five Terms 72
8 Values of ar^ j = 0,2,... ,22 78
9 Values of yk(m), @^(1/3), 0k(l/4),
k = 0,2,... ,22, m = 10 80
10The First n Terms in the Series P (p),
m
m = 10, p= 1/3, 1/4, n = 0,2,... ,22 81
vi


k
C
2
4
6
8
10
12
14
16
18
2 0
22
0
2
4
6
8
10
12
14
16
13
20
22
24
26
88
APPENDIX 1 (Continued)
P
-1/19
-1/2C
-1/21

1.0C00C
1 .cccco
l.CCCCC
1
-0.55028
-C 54 7 77
-C.54549
-c
0.51658
C .50868
C.5C162
c
-C.61415
-C 59436
-0.57696
-0
0.87702
C.82827
0.78648
0
-1.48144
-1.25464
-1.24948
-1
2.95303
2.59C 89
2.30307
2
-6.97997
-5.S1267
-4.93343
-4
19.77901
15.4237C
12.3584C
10
-68.42842
-49.08520
-36.59633
-28
297.92709
191.51C78
130.26340
-571.91135
92
-367
-1/23
- 1/24
-1/25

l.CCCCC
l.CCCCC
l.CCCCC
1
-0.54152
-C 53980
-C.53821
-C
0.48951
C.48429
0.47952
c
-C.54762
-C 52 5 50
-C.52439
-0
0.7187C
C .69C89
0.66622
r\
j
-1.08596
-1.C214C
-C.96543
-o
1.87876
1.71931
1.59483
i
-3.72C57
-2.292CC
-2.94291
-2
8.46292
7.18 757
6. 19143
5
-22.2593C
- 1 7.9 8 5 9 C
-14.8 2 C 64
-12
68.4C097
5 2.04 7 64
40-63786
32
-249.31169
-176.16453
-128.88425
-97
1102.36011
709.76567
479.32931
336
-2132-22218
-1373
1/22
.00000
.54342
.49527
.56155
.75030
.16111
.07017
.25500
. 12928
.16580
.67789
.95599
1/26
.coccc
.53674
.47514
. 51432
.64422
.91650
.47034
.65465
.39946
.41346
.43603
.09741
.70025
.66691


67
sum
. Ps
6 6
X JX ox
a , has been expanded. In fact, the jth derivative
of g(x ,...,xj 0) with respect to x^ is given by
m
6
tfTV g(Xl*--Xm; 0) = Wa StXi>) Vd~)
- t 11 g (x ) ) g(x )
= (-1)J H.(x ) g(x ...,x ; 0),
j a 1 m
where H.(x) is the jth order Hermite polynomial in x. For example,
3
H (x) = 1, H (x) = x 1,
o
H1(x) = x, (x) = x 3x.
It follows that
(5.2.8)
-IV
J \dx/
g(x) dx =
)'
j-1 l
g(x) J
| *
= (-1)J_1 Hj_1(x) g(x)J
= (-1)
H. .(0)
J J-1
y^TT
For even values of j, the (j-l)st order Hermite polynomial vanishes
at the point x=0. Therefore, when expanding the sum, we need to
consider terms that involve only derivatives of odd orders.
6 c
Denoting ^ by o we can represent the kth power of the
a
sum by the multinomial expansion


94
APPENDIX 3 (Continued)
P
n
-1/ 2
-1/ ?
-1/ 4
2
0.16667
C. 1555 1
C.2C578
3
C .04367
C.C6468
9
0.CC578
5
-1/ 5
C.21795
C.07693
0.01672
C. CO 192
-1/ 6
-1/ 7
-1/ 8
-1/ 9
L
0.22335
C 22719
C.23CC5
0.23228
3
C 0 6 5 C 2
C .C5C78
0.05508
C.09842
4
0.3 2 53C
C .03C22
0.03402
C.C3703
c;
C.00487
C .00755
C.CC592
C.01188
6
C.0CC3*
C.CC116
C.CC211
C.G03C4
7
C .CCCC6
C.CCC26
C.C0C55
3
C.CCC01
0.CCC05
9
O.OOOCC
-1/10
-l/n
-1/12
-1/13
2
0.23406
C 23 5 5 1
C.23672
0.23775
3
0.1C105
C 1C 3 2 7
C. 1C5C8
0.10662
4
0.03947
C .04 149
0.04318
C 04462
5
C. 01353
C .01454
0.01614
C.01719
6
0.00385
C .00467
C.CC536
0.00358
7
C.0003 7
C.CC 121
C.CC153
0.C0183
6
0.3C013
C .CCC24
0.CCC35
0.0C046
9
0.0CC01
C CC C C 3
C.CCCC6
C.CC01C
n
e.occcc
C .CGCCC
0.CCC01
C.C0CC2
11
C .COCCC
C.CCCCC
C.COCOO
12
C-CCCOC
C.CCCCC
13
0 oOuCO


k
O
2
A
6
3
10
12
LA
15
let
20
22
C
2
A
fc
8
1C
12
1A
16
18
20
22

2
A
6
8
10
12
1A
16
18
2:
22
o o o r>
APPENDIX 1
VP)
1/ 2
1/ 3
1 .occcc
I.COCCC
c.occcc
-c 175A8
o.occcc
-C .C0 677
O.CCCCv.
-C .C0168
c.cccoc
-C CCC 5 2
o.ococc
-C .C0C21
C.OGCOC
-c .CCC 1C
o.occcc
-C .CCCC6
c. occcc
-C.CCCC3
c.ocooc
-C .CCCC2
0.cccco
-C .OOCC 2
o.occcc
-C.CCCC1
1/ A
1/ 5
l.CCCOC
1.cccco
-0.25671
-C.30772
0.C2771
C.C6523
C.CC2A3
-O.C0299
C.CCCA5
-C.C00A1
C.CCC12
-0.00008
G.CCCOA
-C.C0002
C.CCCC2
-C.CCCC1
C.CCC01
-C.COCCO
C.CCCCC
0.COOCO
o.occcc
-G.COOOO
C.CCCCC
-C.COCCO
1/ 6
1.OCOOC
-0.3AC 1C
C 0 9 7 6 6
-C.01385
O.OCOG2
C.OCCCA
C.CCCO 1
o.occcc
.ocooc
.occc:
0CCGC
.ocooc
1/ 7
1.0CCC0
-C. 36 2 1 1
C 12A83
-C .C269e
C.00 2 A 1
C.CCCC8
C .CCCCG
-C .CCCCC
-C .C0CCG
-C .CCCCC
-c .cccco
-c .occcc
1/ 8
1.CCCCC
-C.38C32
C. 1A757
-C.CAC7C
C.CC6A7
-C.CCC32
-C.CCCC2
-C.CCCCC
-0.CCCCC
-0.CCCCC
-C.CCCCC
-C.CCCCC
1/ 9
i.COCCC
-C.39368
C.16&73
-0.C5A15
C.C1172
-C.C013A
C.CCC02
0.00000
C.COCCO
C.CCCCO
C.GOCOO
0.C0000
1/10
I.COCCC
-C.AGA35
O.18303
-0.96696
C O 17 7 6
-C.9C30C
P.0G02A
C.CCCCC
-C.OCCGC
-C.OGOOC
-o.ocooc
-c.occcc
1/11
1 .CCCCC
-C .A13C7
0.19 7 C A
-C 07895
C .C2A25
-C .CC52A
C CCC69
-C .CCCC3
-C .CCCCC
-L .CCCCC
C .CCCCC
-C.CCCCC
1/12
1.CCCCC
-C.A2C33
C.2C919
-C.C9CC9
C.C3C96
-C.CC796
C.CC1AC
-C.CCCIA
c.cccoc
c.cccoo
G.CCCOC
C.CCCCC
1/13
1.C00CC
-0.A26A7
C.21931
-C.100A0
C.C3771
-C.01107
0.C0239
-C.0003A
C.C0CC2
0.00000
-0.CCCCO
-O.COCCC
85


23
3 (cpy) can be further simplified through integration by parts
and differentiation of (2.3.2). First, integrating by parts, we have
(2.3.3)
i2 2 -i00
3 (cpy) = e^9 Y cos (tcpy)g(t)J
00 )
-^J t cos (tcpy)g(t)dtj-
= y y f t cos (tcpy)g(t)dt.
^ 9y o
Next, differentiating 3 (cpy) with respect to cpy, yields
(2.3.4)
= 9y ^ y ^ sin (tcpy)g(t)dt
d(Vy) Jo
1 2 2 =>
+ e^ y f t cos (tcpy)g( t)dt,
o
where differentiation was permitted inside the integral, since
jt cos (tcpy)g(t)| < tg(t) which is integrable. Combining equations
(2.3.2), (2.3.3), and (2.3.4), we have
~dlv] = Vy 9(cpy) + ~ ^ 7 9y P(f-py)
dGpy)
, 2 2
1 cyp y
V^TT
It follows,
since 3(0) = 0, that
(2.3.4)
yy 1 it2
8 (cpy) = J dt.
o tj2r[
Hence the complex function G(-cpy) can be written as


43
The constants d (m) were evaluated exactly using the recur-
K
rence relation (3.3.4). They are tabulated in Appendix 2 for
ra< 25, k = 0,2 m.
Finally, the probabilities P (p) were evaluated using the
m
formula (3.3.5). Results for p>0 were compared with Ruben's tables
and were found accurate to at least five decimal places. For p <0,
Steck's relation,
Pir/p) = 2 Ik) Pk (l+ (k-2)p) Pm-k (l+kp)
k even
was used in making comparisons. Again, the computed values of P^(p)
were accurate to at least five decimal places. P (p) is tabulated in
m
Appendix 3 for p = , p = 2(1) (25) and 2 < m< 22, and for
p = , p = 2(1)21, and 2 < m < p.
P ~


12
and is written as cp when p <0 with 9 defined as
9 = i=2
a~p
The functions g(y) and G(9y) are defined by
1 Z
g(y) = -i- e oo< y < co
2tt
^0y
Jloo £(t)dt
G(0y) = {
2 2 o
&p"y r~
J_o
p > o
g(t)dt, p < 0
1.4 Summary of the Results of This Dissertation
In the next chapter Lemma 1 and a resuUt connecting the terms
of a binomial series with Tchebycheff-Hermite arad Legendre polynomials
are used to obtain a finite series expression f*or P (p) valid for
r: m
all allowable values of p. For p <0 the resuHts reduce to a work
able form once the real and imaginary parts of G(9y) are isolated.
In Chapter 3 it is shown how the resultes of Chapter 2 can be
programmed to obtain numerical results. Since P^ip) is simpler than
P ip) for computing purposes, a result expresssing P (p) as a sum
r: m r: m
involving P. (o ) j = r,r+l,...,m, is proven. Tfhe accuracy of the
3
computed results is verified by comparison with; exact results for
special values of m and p and with the results >f Ruben and Gupta.


APPENDIXES


69
n\
= P \2)
xm m
(2)
tt
where m will be used to denote the permutation of m elements k
(k)
at a time.
N
For the third term, S T. k. = 2. Since a value of k
3=1 3 3
equal to two would result in an even powered derivative, we need
to consider only values of k. equal to one. Furthermore, these two
J
"ones'- must be assigned to two of the exponents in the quantity
kl k2 kN
(6,5 ) (58) ...(5 6 ) 1 so that the resulting product contains
12 13 m-1 m
four distinct 5's. The first "one can be assigned in N = ways.
kj
Then, there remains f^2) couples, (6i3^i4) containing 6's dis
tinct from the couple (^i^^i^) determined by the assignment of
the first "one.'' Hence, the second "one" can be assigned in
m-2
ways. Since the ones are not distinguishable, there are
/m\/m-2\ 1
\2/V 2/2
possibilities for selecting the two non-zero exponents. It follows
that


TABLE 5
0
1
2
3
4
5
. 50
Zero
Zero
.97862534
2.30187673
.68109061
-14.7500839
. 55
Zero
Zero
1.00864090
2.46469125
1.08317896
-21.1792945
.60
Zero
Zero
1.10872392
3.01034124
2.48383168
-13.8178825
.65
Zero
Zero
1.29975514
4.11498558
5.51922615
-11.4702010
.70
Zero
Zero
1.63001191
6.20062897
11.82707278
-5.4443069
.75
Zero
Zero
2.20215147
10.27259002
25.71608407
9.5845455
. 80
Zero
Zero
3.25336401
19.03426372
60.41653805
41.8041794
. 85
Zero
Zero
5.42986248
41.49665868
168.6658848
71.0033502
. 90
Zero
Zero
11.08343110
121.2957630
683.1599674
-89i.8775674


76
It follows that
(5.3.3) vk(§,ra)
1 r k
¡ [1 -G(§)] kG(§ )
m+k+1 L
1
Since
Yo(§,m) = J* G(u)m dG(u)
o
1
m+1
the Yk(§,m), k = 1,2,..., can be obtained easily from (5.3.3) for
given values of ra and G(§).
The selection of the point G(§) should be made so that the
7^(5,m)'s are small. Since (m+1) G(u)m represents the density on
G(u), we can write
YY(§,m) = Ar E[G(u) -G(§)]k .
'k m+1
Therefore, by letting
G(§) = E[G(u)]
= f (m+1) G(u)m+1 dG(u)
m+1
= m+2
Y-^(§im) equals zero and Y2^,m^ is minimized. However, numerical
work has shown that, although yk(§,m) becomes quite small as k
increases, the values of a.(§), j = 0,1,..., corresponding to the
J
ITH-1
point § that satisfies G(§) = become exceedingly large.
(For example, with m=8, § = 1.28155157 and c*10(§) = -3,825,025.96.)


ACKNO WLEDGMENT3
I would like to express my appreciation to my major professor,
Dr. John Saw, who suggested the topic of this dissertation, and who
was always available for assistance. Appreciation is expressed also
to the other members of my supervisory committee, Professors R. L.
Scheaffer, P. V. Rao, and Z. R. Pop Stojanovic.
Also, I would like to extend my thanks to the other members,
faculty, students, and staff of the Department of Statistics. They
made my stay at the University of Florida both rewarding and enjoyable.
The manuscript was typed by Mrs. Edna Larrick. Her patience
and assistance through those final pre-deadline weeks will always be
remembered and appreciated. Also, I would like to thank Mrs. Deborah
Ingram for her excellent work drawing graphs.
Finally, I express deep appreciation to Professor Paul Benson
of Bucknell University. Without his guidance and encouragement I
would never have entered the field of statistics.
iii


29
Now using (3.1.1) we find that
x 3.-1 1
J2(p) = sin P 2
Finally, substituting the above results into Lemma 2, we have, for m=2,
and for m = 3,
P0;2 > 1 1 -1
P1:2 P2:2

=I+SS1,,'lp'
, 1 3.-1
P0:3 -1
Pl:3 3 3-1
P2:3 1 3.-1
P3:3,P) = 8 + 4S S1" P-
Notice in the above results that
P (p) = P (p),
r:m m-r:m
and
E P (p) = 1.
_ r: m
r=0
These properties also hold in general. The first result follows imme
diately by letting r = m-r in Lemma 1, page 11, and noting that
G(8y) = l-G(-9y) for both positive and negative rho. Lemma 2 is used
m
in showing the second property. Since E c (r,m)=0 for k=l,2,...,
r=0 k
we have


37
Figure 4.
i
The function (-h)5 1^ [f(u)]
p = 1/2, p = 1/10.


41
In proving (3.3.3), we begin with the basic definition of
P (p). That is,
r: m
P
r: m
(P)
() P(X1>0 V; Xr+1<0'
-X,
m
Next, let A. be the event [X.>0] and denote its complement by A.
3 3 3
Also, define I as the intersection A A .
K J. 2
. A, Then P (p) can
k r: m
be written as
P (p) = i"1) PyA . .A A ... A
r:m \r/ \ 1 r r+1 m/
= (m) p(l -A AA ... A ^
\r/\r 1 rr+1 m /
Applying de Morgan's rules, and since P(I ) = P (p), we have
r r
P (p) = j P (p) -p(a ...A U A
r:mr \r/ L r \ 1 r r+J/J
J-1
= 0[pr(P)-pCu"A1...ArAr+J)] .
j ^
Finally, using the formula for the probability of a union, we have the
desired result. That is,
P
r: m
(0)
(HV -L^KV-AA+j)
3 = 1
i < i
J1 J2
5(A1
-EE PA....A A .A
r r+3 r+j2
+ E E E p( A .
i < i < i v 1
J1 J2 J3
.A A .A .A
r r+Ji r+j2 r+j3


74
The function
(u) j
0,1,., is given by
a. (u)
J
_L ( d x J
j: \dG(u)/
g(u) ,
and can be written
The function
rpu)
LVdG(u)J
j
g(u)^
- g(u)J(j+l): a (u)
is tabulated by Saw (1958) for j = 0,1,...,10.
Substituting the expansion (5.3.2) of g(u) into (5.3.1),
we have
P (p)
ra
= (2n)
h-1
2
i
h2
J
E
j=0
Qf. (§) [G(u)
*3
y
G(§)]Jj
h-1
G(u)m dG(u)
h-1
= (2n) 2
o k=0
Pk(§;p)[G(u) G(§)]kG(u)mdG(u),
where the constant 3, (§;p), k = 0,1,..., can be determined from the
Q?k(§) s. Assuming that an interchange of integration and summation
is permitted, we have


m
o
3
4
5
6
7
6
9
1C
11
12
13
14
15
16
17
18
19
20
21
22
2
3
4
5
6
7
3
9
10
11
12
13
14
15
16
17
18
19
20
21
22
o o o o
92
APPENDIX 3 (Continued)
1/10
.26594
. 1489 1
.08709
.05286
0.01413
0.00955
C.0C655
C.00462
C.0C33C
0.00239
C.00 175
C. 3C13C
C. 0009E
O.OCC74
0.0C057
0.0C044
0.0CC34
c.00027
0.0CC2 2
P
1/11
C .26449
C 14673
C.08480
C.05C77
C.03135
C.01990
C.01294
C.C0S6C
C.00583
C.C04C2
C.00282
C.0C2C 1
C.00145
C.00 1C 6
C.CCC78
C.COC 58
C .CCC44
C .CCC34
C CCC2 6
C .CCC2C
C.CCC16
1/12
0.26328
C. 14492
C.08289
C.C49C4
C.02989
C.01870
C.01198
0.CC784
C.GC524
C.C0356
C.C0245
0.CC172
C.CC122
C.CCC83
C.CC064
0.CCC47
C.CCC35
C.CCC26
C.CCC2C
C.CCC15
C.CCC12
1/13
0.26225
0.14336
C.08129
0.04759
0.02867
0.01772
C.01120
0.00724
G.C0476
C.00319
C.C0217
0.0015C
C.C01C5
C.G074
0.00053
0.G0039
C.C0Q28
G.C0021
C. C0C16
C.C0012
C.00CC9
1/14
1/15
1/16
1/17
0.26138
C.26C62
C. 25995
C.25937
0.1420?
C 14C93
0.13993
C.13905
C.07992
C.C7874
C. 07 771
0.07680
0.04636
C .04530
C.04438
C. 04357
0.02764
C .026 76
C.C26CC
C.02533
0.01669
C.01619
C.C1559
C.C15C6
0.01056
C.C1CC1
C.00954
C.C0914
C.CC673
C .CC631
C.C0596
0.00566
C.0C438
C .00406
C.00379
0.00356
C.00289
C.00265
C.C0245
C.00228
0.00194
C .00176
C.CC161
0.00148
O.OOI32
C.00118
0.C0107
C.00098
0.0C091
c .cocei
C.CCC72
C.00065
C.0CC64
C .C0C56
C. CCC49
C.G0C44
0.00045
C.COC 39
C.CCC34
C.C0G30
C.0C032
C.00 C 2 8
C.CQC24
C.00021
O.OC022
C .0CC2G
C.CCC17
G.G0C15
C.CCC17
C .CCC14
0.CCC12
C.COCIO
0.00013
C .COCIO
C.CCC09
C.CCG07
0.0CCC9
C CCC C 8
C.COCO6
C.00005
C.CC0C7
C .CGCC6
C.CCC05
C.GCC04


52
represent the standardized distribution function of Y^^- Y, we can
write
(4.3.1)
P (Y < r) h P(Y, x Y < 0)
n (V)
' Y
= P
(V) Y "
l^V
^ /
2 V '
= F.
U
l^V
5* I
2^V
where ¡_l and § represent the mean and variance of Y Y,
1 V 2 V (v)
respectively. An approximation to the distribution Fv(x) can be
obtained by using Edgeworth's expansion (see, for example, Cramer,
p. 229). Letting § represent the kth central moment of Y ,-Y,
we have, using the first four moments,
(4.3.2)
Fv(x)
= G(x) g (x)
1 3 V
6
-3/2
V
2
(x 1)
1
(4§V
, 3
+ 24
T2
2 v
(x 3x)
where G(x) and g(x) are the standard normal distribution and density
functions, respectively. Thus letting x
A
in equation (4.3.2),
we can approximate P^(Y ^ r).
By using a power series representation, the moments, § ,
k V
k = 2,3,4, can be determined. Saw (1958) has shown that the kth
moment, of the Vth order statistic can be expressed as


73
5.3 A Series Resulting ffom an Inverse
Taylor Series Expansion of g(u)
For oositive rho, consider the expansion for P (p) given
r: m
in Lemma 1:
CO
P (p) = (m) f [1 G(9y)]r [G(0y)]m r g(y)dy
r: m \r / _
2 -1
where 9' = p(l-p) Since P (p) = P (p) we can write
r: m mr: m
P (p ) = P (p ) so that
m u: m
P (p) = f G(8y)m g(y)dy
m
After the change of variables, u = 8y and h = 1/0 we have
2 1
=o t 1
1 a ml
(5.3.1) P
~iu ~2
m(p) = G(u)1" -L. e 6 dy
*J2rr
h-1
2 i
= (2tt) hs f G(u)m g(u)h 1 dG(u) .
By expanding the density, g(u), in an inverse Taylor's series, the
integrand will contain only terms involving G(u). Expanding about
an arbitrary point G(§), we have
(5.3.2)
g(u)
I
[G(u) G(§)]J ( d \j
VdGJ g(U)'
J=
u=S
= £ Of (§)[G(u) G(§)]J .
j=0


66
Finally, addition of equations (5.2.4) and (5.2.6) gives
a> cd
g(xi Vp)
CO CO
p=0
£ xm0)J
a < b z b
m
i. E t .x.
j=l J J H
e dx_
.dx = 0,
m
which implies that
Vg p =0 a Since derivatives of all orders exist and are continuous, the preced
ing process can be repeated any number of times. Hence, the identity
(5.2.2) is proven.
Using this identity in the Maclaurin expansion of g(x,...,x ;p),
1 m
given by (5.2.1), and substituting the resulting expression into P (p ),
m
we have
(5.2.7)
P (P)
m
-J-/ l
k=0
a < b
£-) v
b
.,x ;0)dx_ . dx
m 1 m
E E
a < b
,x ;0)dx.. dx .
m 1 m
The utility of this expression for P (p) depends on how readily each
m
term in the series can be determined and on how quickly the series
converges. Differentiation and integration are no problem once the


54
= E (-1)
£=0
t a21
(ty
X! (2n) j=0
j "v j:
= E E J
=0 j=0 X! j (2n)
j
After the change of variable, k = 2 + j, it follows that
00 # k
cp (t) = E Ut)
k=0
k'.
k!
"k/2
E (-r/
_£=0 x: (k-2X>: (2n)
1 (k-2£)^V
]
Therefore,
- k k/2 i k;
E(Y(vfY) = n + ^ (-1) j (k-2£)^V
( ; k 1=1 X! (k-2X)! (2n) (
and, in particular,
E(Y(V)- Y) = A
E(Y(V)- Y) 2^V n
E(Y(V) Y) 3% n l^V
E(Y(V)-Y) n 2^ + 2
n
Using this information, the central moments, jc§v of Y can be
expressed as functions of u For example, with k = 2,
K V
5v EV'
(V)
= E(Y(V)-Y)2 2E(Y(V)-Y) A +
1 2 2
= 2^ 21^V + l^v
2 1
2^ l^V n


5
Substituting for v in the above equations, we have
aj
(1.2.5) p 2 a q ^QQ ~ 1,
j/B jB BB
p E a Q + (l-o ) a q + pQipn ^ 0.
i*t oB aB BB
Subtracting these last two equations, we have
(1,2,6) a90 ao& ~ l-p
It is clear from the preceding equations that a^g is independent of
a and 8. Hence, equation (1.2.5) simplifies to
(1.2.7) (m-l)pa^ + a^ = 1.
Finally, the solutions of equations (1.2.6) and (1.2.7) are
06 (1 p)[1+(m-l)p J
1+ (m-2)o
88 (l-p ) [ 1+ (rn-l)p ]
-1
as was to be shown.
Using the scalar representations of J V( and x V x in the
density function, (1.2.1) may be written as
m-1
(1.2. s) Pr;ri m m-l 1
(2n) 2 (l-p) 2 [l+(m-l)p ] 2
[r, m]
exp
m 2
[l+(m-2)p] E x. 2p E E x.x.
i=l 1 i 2 (l-p )[l+(m-l)p]
dx . dx
1 m
Unfortunately, although void of matrix notation, the above representation
of P (p) is not too desirable for obtaining numerical results. A more
r: m
workable form of P (p) is needed.
r: m


63
1 y 2
f 2 J
g(x ,...,x ; O) = (2tt) 2 e ;i~1
1 m
_ a= J
we could simplify the integrand by expanding g(x ^...x^; p) in a
Maclaurin series in rho. We have
oo k
0 ** { £
<5-2-1) e A et W g k=0
|p=0
Unfortunately, as a function of rho, g(x,...,x ; p) is quite compli-
1 m
cated and it is not feasible to take derivatives with respect to rho.
However,.the following identity simplifies the problem to. some extent:
(5.2.2)
$57 g(Xi--'Xm; P)
P 0
= (s s
i 6 6
g(x- ,.. ,x ; 0) .
x. 6x.) m'
i J
In proving identity (5.2.2), we use the characteristic function,
cp(t ...,t ), of the variables X ...,X By definition
m
m
(5.2.3)
m
9(tr.
.t)
m
= E e
i. Z t.x.
J-l J
CO CO
J
m
E t.x.
j=i J J ,
e g(xl(
.,x ; p)dx...dx
m r 1 m
00 00
4 t'vt
e
= e
-l \ t2
2 >i J
p E t.t,
i


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87
APPENDIX 1 (Continued)
k
-1/ 2
-1/ 4
P
-1/ 5
-1/ 6
C
l.CCCOC
1 .ccccc
1.CCCCC
l.COOCC
2
-0.62452
-C .74129
-C.69226
-C.65990
4
2.44596
1.51C71
1.15617
C.96895
6
4.C4 66 6
-2.53547
-1/ 7
-1/ £
-1/ 9
-1/1C
1.0CC0C
1 .CCCCC
l.CCCGC
l.COCCO
2
-0.63689
-C .61969
-0.60632
-C.59565
4
0.35236
C 775C5
C.71856
0.67592
6
-1.86571
-1.49315
-1.2586C
-1.09863
3
7.0626C
4.46698
3.2C847
2.48636
10
-12.72389
-8.09368
-1/1 1
-1/12
-1/13
-1/14
0
1.CCCCC
1 .CCCCC
l.CCCGC
l.COCCO
<.
-0.53692
-C. 57967
-0.57353
-0.56827
4
C.64262
C .61592
0.594C3
C.57576
6
-0.98317
-0.69624
-0.82862
-C.77462
8
2.3272C
1.71284
1.46657
1.32C03
1C
-5.71S9C
-4.32794
-2.43337
-2.63220
12
23.39377
14.93534
10.42892
7.75036
14
-43.61756
-27.917C0
-1/15
-1/16
-1/17
-1/18
0
l.CCCOC
1 .CCCCC
l.CCCOC
l.COCCO
2
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-C.55303
4
C.56029
C 547C3
0.53553
0.52546
6
-0.72056
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-0.66318
-0.63687
8
1. 18986
1.08676
1.CC23C
0.93454
1C
-2.39976
-2.C7664
-1.83C32
-1.63672
12
6.03114
4.86226
4.C3 1C 7
3.41332
14
-19.31984
-14.15326
-10.82916
-9.57463
16
62.16048
52.68241
36.20390
26.21621
18
-155.97371
-100.15185


36
Letting f (u) denote the (jj-l)st term of the sum, we have the recurrence
relation
f (u) = u
v/2
u 2 i 1.
f .(u) = %- ,"-J. f .(u) .
J 2 j(2j + l) j-l
j = 1,2,...,
which can be easily programmed.
Although the Legendre polynomial has the imaginary argument,
f(u), for even values of k it is a real function. Hence, for comput
ing purposes, we can avoid complex numbers by defining the function
*
L (-t) by the recurrence relation,
K
L*Ut) = 1
o
(3.2. 1) L*(-t) = -t
vZt) = (-i)k irt LhUt) ¥ k*2-
Then the function L, (.t) can be determined by
k
L*at}
Lka*> = .
-iLkat)
k even
k odd.
The above relation can be verified by substituting into the recurrence
relation (3.2.1) and comparing the results with the recurrence relation
for L^it) given by (2.1.5).
Once the Legendre function, Lk[i.f(u)], is evaluated, we can
A r_, 2-,-h-l
, (_h)" T r .-f/nM l ~ U
examine the integrand,
Lk[f(u)] j_e j of JR(p). The
integrand is plotted in Figure 4 for p = k = 2, and for
p = k = 2 and k=10. Notice that the scale of the graph differs
from those in Figures 1, 2, and 3, since,for large k,|Jk(p)f is quite


61
TABLE 6
MOMENTS OF Y Y, n = 19
r
*
V
**
A
§ t
2^V
§ +
3 v
X
§ l
4bV
1
18
1.37994
.11015
.01897
.04152
2
17
1.C9945
.07308
.00868
.01768
3
16
.88586
.05484
.00492
.00975
4
15
.70661
.04416
.00308
.00624
5
14
.54771
.03739
.00202
.00442
6
13
.40164
.03298
.00131
.00342
7
12
.26374
.03020
.00079
.00285
8
11
.13072
.02866
.00038
.00255
9
10
.00000
.02816
.OOOOO
.00246
*
**
t
*
II
II II
n r .
W.
E(Y(v>- 7 -
i\)k k =
2,3,4.


LIST OF FIGURES
Figure
Page
i
1 The function L^[2G(u) 1]
2-ih-l
[e--],
p = 1/10
32
2 The function h^ L, [2G(u) 1]
k
2-,h-l
, p = 1/2
33
p 2 2-ih-l
-?u
3 The function h L^[2G(u) -1] J^e 2 j p = 9/10
34
? r* 2-n -h-i
4 The function (-h) ^[1(^)1 I e U j
p = 1/2, p = -1/10
37
vii


71
3! /m\/m-2\/m-4\ 1
00 00
:ym-4> l n f 0
lllTll \2 A 2 A 2 J 3T Hx
c
'l
-X Ox Ox Ox ox
00123456
g(x ,...,x ;0)dx ...dx
m
V
3
P_
3!
(2) f H2(0)12 fl\m 2
2 L vss-
+ m(4) |_-
r H (0)i H (0)-, 3 m-4
.1 r oHJ m
J L rr J \2 /
v^rr ^ ~ a/2tt
m(6) r Ho(0)l6
,3 L
J
\2j
p3^l^mrra(2) 4m(4) m(6)l
= 3: ^ ^ " n2 + r3 J
Finally, for k=4, values of the k.'s equal to three and
J
"one" or "two" and two "ones" can both be assigned to form a product
3 3
having the form i^2^i3^i4- An assiSnment of four "ones can
3
result in either a i^^ig^i^i^ig product or a 6j_ 52...5ig
product. It follows that
-{fe -
r_iL_ r
L2:i!i: \:
m^
2)
[2(m-2)](m
r H (0 ) -> 2 ¡ H (0) , 2 ,,Nm-4
l"!r A '
-_ j li)
vss- v2/
. f 4! l]r"2(V Ho(0,l5
(T)
\2 J
m-6


2
3
4
5
6
7
8
9
1C
11
12
13
14
15
16
17
2
3
4
5
6
7
9
9
1C
11
12
13
14
15
16
17
18
19
2C
21
o r>
95
APPENDIX 3(Continued)
-1/14
-1/15
0.23862
C .23938
0. 1C793
C 1C5C7
C. 04586
C.04694
C.018 1C
C .01891
C.00654
C.CC7C4
0.00212
C .00 2 39
C. CC06C
C.CO C 72
C.0C014
C .CCL 19
0.00003
C.C0CC4
c.occcc
C .0CCC1
C.CCCCC
C.CCCCC
c.occcc
C.CCCCC
C.OCOGC
C.CCCCC
C .CCCCO
-1/16
-1/17
0.24CC5
0.24063
0.11CC7
C.11095
C.C4789
C.04873
C.01962
C.C2C26
C.CC749
G.CC790
C.CC263
C.CQ286
C.CCC34
C.C0C95
C.CCC24
C.CC028
0 C C C u 6
C.C0C07
C.CCC01
C.C0CC2
C.CCCCC
C.CCCCO
C.CCCOC
C.CCCCO
O.CCCOC
o.coocc
C.CCCCC
0.CCCCO
C.CCCCC
C.CCCCC
c.coccc
-1/18
-1/19
-1/20
-1/21
0.24115
C.24 162
C.242C4
C.24242
0.11173
C.11243
C. 11206
C. 11363
0.04948
C.05C16
0.C5C76
0.05131
C 028 3
C.02124
C.C2181
C.C2223
0.0C827
C .00861
C.CCS92
C.00920
C.0C3C 7
C .CC227
C.CC345
C.C0361
C.00105
C .00116
C.CC125
C.C0134
O.OCC33
C .CCC38
C.CCC42
C.C0C46
C.COOCG
C.CCC11
C.CCC13
C.C0015
C.CC0C2
C .CCCC3
C.CCC04
C.CCC04
c.cccic:
C .0CCC1
C.CCC01
0.COCCI
C.CCCCC
C.COCCC
C.CCCCC
C.OOCCC
C.CCCCC
C.CCCCC
C.CCCCC
C.COCCC
C.OCCCC
C C 0 C u C
C.CCCCO
C.COCCO
c.occcc
C.CCCCC
C.CCCCC
C.COOCG
c.ocooc
C.CCCCC
C. CCCCO
C.COCCO
o.ococc
C .CCCCO
C.CCCCC
O.OCCCO
C.CCCCC
C.CCCCC
C.COCCO
C.COCOO
C.COCCO
C.CCCCC


56
C3(V2) = H3(V2) 2Ho(PV1) H3(PV1)
- 2H1(PV,1) H2(PV,D 4,
^V1 2,o - 2H1(PV,1) H3(pv>1) 8 ,
C5(PV2) = W2> 2Ho(PV1) W1*
- 2H1(pv,l) H4(pvl) 2H2(pv,l) H3(pv,1) 16.
The constants, C.(p ,3) and C.(p ,4), can be found in the Same tedious
J J y
manner. The values of Cj(pv>k), k = 2,3,4, j< 5, and
= .50, .55 90 are tabulated in Tables 3, 4, and 5.
Using these tables, we can approximate the moments 2§^, 3V
and 4§v and then use relation (4.3.2), with x = in order
§ 2
2^ V
to obtain an approximate distribution on Y. As an example, we take
n = 19. Then
V
PV = n+T
19-r
20 *
Tables 3, 4, and 5 were used to approximate 2§^, 3^v ancl 4^v
for r = 1,2,... ,9. The results are listed in Table 6. The values for
were taken from tables computed by Teichroew (1956). Teichroew's
tables were also used to check the accuracy of the series approximation
for § For n = 19, the approximation was accurate to five decimal
places.


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
dr"
J. G. Saw, Chairman
Professor of Statistics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Assistant Professor of Statistics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
P. V. Rao
Associate Professor of
Statistics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Z. R. Pop Stoj anovic /
Associate Professor of Mathematics


m
2
2
4
5
6
7
3
9
10
1 1
12
13
1*.
15
16
17
13
19
20
21
22
2
3
4
5
6
7
o
9
10
11
12
13
14
15
16
17
13
19
2Q
21
22
o o o
APPENDIX 3
1/ 2
P (p)
ci
1/
1/4 1/5
0.33333
C.25CCC
0.2CC0C
0.1666 7
0. 14286
O. 125CC
O.1111 1
0.1CC0C
G.09091
C.08333
C. 07692
0.07143
0.06667
0.0625C
C.05882
.05556
.05263
.05CGC
C.04762
0.04545
C.04348
C.304C9
G.2C613
C 14974
C 11413
C .C9C12
C .07 31 1
C .06C61
C.C5113
C .C4375
0 .C3 790
C C 3 3 1 8
C .0253C
C .C26C8
C .02 338
C C 2 1C 8
C .01912
C .01742
C .01594
C .01465
C .01351
C.C1251
C.29C22
C.18532
C.12648
C.C9C66
0.06748
0.C5176
0.C4C67
C.C3262
0.0266C
C.02-202
C.C1845
C.01564
0.01338
C.01155
C.C1C04
C.CG879
C.CC775
C.CC687
C C C 61 2
C.CC548
0.C0492
C.282C5
C.17307
C.11301
C.G7741
G.05508
C.04G43
C.03044
C.C2343
0.01836
C.01463
G.C1132
0.C0967
0.00799
.C0668
0.C0563
G.CC478
0.00409
C.C0352
C.C0305
C.00266
0.C0233
1/ 6
1/ 7
1/ 8
1/ 9
".27665
C 2728 1
0.26995
0.26772
0.16498
0.15922
C.15492
0.15158
0.10422
C.096C4
C.C9345
C.08990
C.06893
C.06 3C6
C.05875
G.05546
0.04733
C.04205
C.C3825
C.0353S
C 0 3 3 5 2
C.02892
C.02566
0.C2323
0.02437
C.C2C42
0.C1766
C.01565
r .01812
C 0 1 4 7 4
C.C1244
0.C1079
0.01374
C .01086
C.CC694
C.C075S
0.01059
C .00814
C.CC654
.00543
0.00929
0.0C62C
C.CC486
C.C0395
C.0C655
C .00479
C.C0366
C.C0291
0.00528
C.CC374
C.CC28C
C.C0216
0.00428
C .00296
0.00216
C.00 i 65
0.0C35 1
C .00237
C.C0169
C.00126
0.0C29C
C .00191
C.C0133
C.00093
C.00242
C .00 155
C.CC1C6
C.C0G76
0.0C2C3
C.C127
C.CC085
0.C0C60
0.00172
C.00105
C.CCC69
C.C0C48
0.CC146
C.00088
0 C C 05 6
C.C0038
O.OC125
C.CGC73
C.CCC46
0.CC031
91


Since the last sum equals zero for all r and m and since b^(m) c^(r,m)
does not equal zero, J (^) = 0.
3.2 Evaluating the Integral J (p)
K
The Case When Rho Is Positive
Recall from Lemma 2, with p = (h+1) ^ > 0, that
1
dG(u) .
Before attempting numerical integration, the integrand should first be
investigated for different values of k and p For given values of
G(u), the Legendre polynomials can be evaluated from the recurrence
relation (2.1.5). Also, the value of u corresponding to G(u) is given
to eight decimal places in The Kelley Statistical Tables. For each of
11 9
the cases p = and the integrand is plotted in
XU Z XU
Figures 1, 2, and 3, respectively, for both k = 2 and k=10.
Since numerical integration is most accurate when the function
being integrated is well behaved, we should expect good results for
p< ~ with the accuracy increasing as rho decreases. Also, since k
dt
is the order of the polynomial being integrated, the better results
should occur when k is small. When p > the integrand approaches
infinity as G(u) approaches one as can be seen from Figure 3. In this
case, it is not likely that numerical integration will give accurate
results.
Using Simpson's rule with 200 intervals, the interval width
- .0025, with G(u) taking the values .5 + (,0025)j,
equals


22
to simplify J (p) the real and imaginary parts of GCcpy) must be
A
isolated. Denote these real and imaginary parts by oKcpy) and 0 (cpy) ,
respectively. Then G( GCcpy) = aCpy) + 3(cpy) ,
(2.3.1)
G(-cpy) = o(9y) -0(cpy).
Using the definition of GC>Py) given in equation (1.3.2), we have
2a(cpy)
o
g(t)dt + J e^1^7 g(t)dtj-
_ CO
= e
i 2 2
9 y
-tcpy
g(t) dt
i 2 2 i 2 2
= eS9 y y
= l.
Therefore,
oK9y) = g
Subtracting equations (2.3.1) gives
2i6py) >2y2 {J e^yt g(t)dt j ?yt g(t)dt}
= e
9
2 2
P f j
^tcpy ^tcpy'j t>
.tcpy fcpy
Since e e = 2^. sin (tcpy) we have
i 2 2
3 (9y) = ^y ;
(2.3.2)
sin (tcpy) g(t)dt.


TABLE 4
Cj(Pv.3>
-X
0
1
2
3
4
5
. 50
Zero
Zero
Zero
Zero
Zero
Zero
. 55
Zero
Zero
.14261956
.46342873
.77469507
.2860125
. 60
Zero
Zero
.30026404
.99445568
1.67177376
. 47107 49
. 65
Zero
Zero
.49242777
1.68698946
2.86385938
.3195549
.70
Zero
Zero
.75034258
2.70841421
4.66613950
-.8926114
. 75
Zero
Zero
1.13308641
4.41851254
7.77768619
-5.7261556
. 80
Zero
Zero
1.77168120
7.74697987
14.08664124
-25.1106438
. 85
Zero
Zero
3.02054891
15.75323340
30.27071486
-122.3376979
. 90
Zero
Zero
6.18793059
43.35945955
93.78561212
-932.6502938


3
J J J '
X > O X.
1 1
i < r i > r
= f S(xi-x2 V dXl>dX2 dV
[r ,m]
where [r,in] will be used to denote the range of integration
fx >0: 1 < i < r; x <0: r+1 < i < m] .
1 i i J
In the case r = m, we will, for convenience, write
K < o
/ /m\
(1.2.1) F (p) = )
r:m r \r/
P (p ) rather than P (p ) .
A first approach to the problem might be to write down the
density function g(x ,x ,...,x ). From multivariate theory we have
1 £ m
m 11 ./ -1
2 | i 2 2 V -
(1.2.2) g(xi,x2 xm) = (2tt) j V| e
_ 00 < X
i
i = 1,2,...,m,
where x denotes the vector (x ,x ,...,x ) and V is the dispersion
matrix given by
1 2
n p
p i
m
V =
Due to the simplicity of the dispersion matrix, the determinant |VJ
is easily evaluated and the quadratic form x V 1 x has a simple
scalar representation. In fact, it will be shown that
(1.2.3) |V| = (1 -p)m"1[l+(m-l)p],


50
Y needed to form the rejection region can be determined using the
approximation. With a 10% level of significance and a two-tailed
alternative, the critical values are the solutions to the equation
1 n
r + 2 ~ 2
= 1.645 .
VVar (Y)
(Notice that i was added as a correction for continuity factor.)
As an example, with n = 19, we have
or
+ 2
1.31320
19
2
1.645
r = 9 2.1
= 7, 11.
As a check, from the small sample distribution on Y given in Appendix 4,
we have
p1Q = P10(Y^ 11) = .0595.
y iy
Table 2 compares the small sample distribution of Y with the normally
approximated distribution for n=19. (The results of the approxima
tion which will be discussed next arelisted in the third column.)
We would expect the approximated results to increase in accuracy as
the sample sice increases.
An alternate approach to the problem of approximating the null
distribution on Y is through the use of order statistics. Consider
again the random sample of standardized variables, Y,Y_,...,Y Then,
12 n
letting Y^ denote the vth largest order statistic, the events [Y< r]
and [Y^ < Y] are equivalent if we set V equal to n r. Letting F^(x)


60
These moments were used in the expression for F (x) given by
(4.3.2) to obtain an approximate distribution on Y for n--19.
The results are given in Table 2, page 51. For n= 19, there appears
to be little difference in accuracy between the two approximating
methodscertainly not enough to justify the extra labor involved in
computations for the latter method. However, the second method does
work well and is at least of theoretical interest.


38
large. Also, unlike the case when rho is positive, the functions do
not cross the G(u) axis. These differences, together with the smooth
ness of the curves, should make numerical integration even more accu
rate for p < 0.
3.3 Computing P (p )
m
Once the J^ip), k = 0,2 m, are computed, Pr.m(p) is deter
mined, for r = 0,1,...,m, by the expression giveai in Lemma 2. That is,
m
(3.3.1) P (p) = E b, (m) c (r,m) J (p) .
r:m r n k k k
k=0
k even
Unfortunately, since the Tchebycheff-Hermite polynomial, c (r,m),
K
cannot be expressed in a recurrence relation, it is difficult to compute
P (p) using the above expression, except for snnall values of m.
r: m
However, with r = m, c (r,m) simplifies to, say c (m) ,
K
(3.3.2)
, m!(k!)2
Vm) (m-k) I (2k)! '
Furthermore, the probability P (p) can be written as a linear combi-
r: m
nation of P^ip), k = r,rt-l m. That is,
(3.3.3)
r: m
(P)
-C)
m-r
E
j=0
(-1)'
m-r
r+J
(P>
(ITie proofs of (3.3.2) and (3.3.3) are given at tdie end of this section.)
Hence, we use (3.3.1) only in the special case wh>en r = m. Relation
(3.3.3) then can be used for computing P (p) wihen r ^ m.
r: m


86
APPENDIX 1 (Continued)
k
1/14
p
1/15
1/16
1/17
0
1.00G0C
1 .ccccc
1.ccccc
l.COGOO
2
-0.43173
-C .43629
-C.44C23
-C.44350
4
0.22917
C. 23749
C.24491
0.25159
6
-n.1C992
-C. 11872
-C. 12686
-C.13439
3
C.0444C
C .05C96
C.Q5733
C.C6348
10
-C. 01447
-C .C16C9
-C* C2185
-C.02570
12
O.OC363
C .C0512
C-C0682
C.C087C
14
-C.CGC66
-C.J011C
-C.CC168
-C.C0239
16
0.3CC07
C.C0C17
C.CCC31
C.C051
18
-c.ooccc
-C .CCCC1
-C.CCCC4
-C.C0C08
2C
-o.occoc
c .coccc
C-CCCOG
C t o u 01
22
_ p r* n r> n
L V v O G
c. ccccc
-c.cccoc
-o.cooco
1/18
1/19
1/2C
1/21
c
l.PCOOC
l.OCCCC
i-CCCOO
l.OOCCO
2
-C.44692
-c.44972
-0*45223
-C.45451
4
0.2576 1
C.263C6
0* 26807
0.27263
6
-0.14136
-C.14737
-C. 1539C
-C.15953
8
0.0694 1
C.07510
C C 8 C 5 5
0.08577
1C
-C.02961
-C.C3352
-0*03742
-C.4129
12
0.0 1C 73
C.01289
0*01516
C.01751
14
-C.0C323
-C.C0419
-C.CC526
-C.00644
16
0.0CC76
C .00112
C-CC154
C.002C2
13
-c.rcni5
-C.CCC24
-C CCC3 7
-C.C0053
20
O.CCOU2
C.CCCC4
0.CCCC7
C.C0G11
2 2
-o.ocooc
C l 3 u C C
-C-CCC01
-C.CC0C2
1/22
1/23
1/24
1/25
n
w
1.CCCCC
1.CCCCC
L-CCCOC
l.OOCCO
2
-0.45655
-C .45847
-C* 46C2C
-C.46179
4
C. 27682
C 2 8 C 6 8
C.28426
C.28757
6
-0.16479
-C 1697 1
-C* 1 7433
-C.17866
3
0.99C76
C .09 5 53
C* 1CCCS
0.10445
10
-C.0451C
-C .04665
-C*C5253
-0.05613
12
0.0199 1
C .C2237
C.C2485
C.02 73 5
14
-C.0C77 1
-C.CC8C6
-C* C1C48
-0.01196
16
C.0C256
C .CC22C
CmCC39C
0.C0465
18
-0.3CC73
-C.CCC98
-C.CC126
-0.00159
20
0.00017
C.0CC25
C.CCC35
C.C3G47
22
-0.CCC03
-C.0CCC5
-C.CCC08
-0.CC012


83
I
(h-1)(h-2)
(m+2)(m+3).
(h-k)
(m+k+1)
I
Hk(u) g(u)h k 1f(u;m+k+l)du,
where H (u) is the kth order Hermite polynomial in u. Therefore,
k
after h-1 integrations, we have
(5.4.1)
P (p)
m
h-1
(2tt) 2 h^(h-l) (h-2)...2-
(nw-1) (m+2). . (m+h)
-1/ W)
_CO
f(u;m+h)du
E [R , where U, is the largest order statistic in a normal sample of
(m+h)
size m+h.
For example, with p = 1/3 and p = 1/4, expression (5.4.1)
simplifies to
and
P (1/3)
m
2rr
(m+1)(m+2)
E U
U(m+2)
P d/4) = T T//^9 w oV r E C2 l"! ,
m (m+1) (m+2) (m+3) J
U(m+3)
respectively. Using the table of moments of extreme order statistics
computed by Ruben (1954), expression (5.4.1) can be used to determine
P^ip) for values of m and p, satisfying p S 1/12 and m + p 1<51,
where p 1 is a positive integer.


Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
THE PROBABILITY THAT PART OF A SET OF EQUICORRELATED
NORMAL VARIABLES ARE POSITIVE
By
Thomas Ray Hoffman
March, 1972
Chairman: Dr. J. G. Saw
Major Department: Statistics
The probability that part of a set of equicorrelated normal
variables are positive is defined by a multiple integral expression
involving the multivariate normal density function. Although much
research related to this integral expression has been published, most
results do not include a practical method of its evaluation. Also,
when the correlation is negative, no direct method of evaluating the
integral expression is available. In this paper we discuss several
methods of expressing the integral. One of these expressions, valid
for both positive and negative correlation, is used to obtain numer
ical results.
A transformation is used to simplify the integral expression
for the probability that part of a set of equicrrelated normal var
iable are positive. Then the probability can be written as an integral
involving the real normal distribution function when the correlation
is positive, and the complex normal distribution function when the
correlation is negative. For positive correlation, this integral
expression has been used by other authors to obtain numerical results.


2
been obtained when the correlation is negative. Only two authors,
Ruben (1954) and Gupta (1963a) give numerical results.
It is the purpose of this dissertation to find at least one
method of evaluating the multivariate normal integral that works for
both positive and negative correlation and that can be used easily to
obtain numerical results. Then the small sample distribution of David's
statistic can be given as an application to the more general problem.
1.2 Definition of P (p)
r: m
Suppose the m variates X,X-.,... ,X each with zero mean and
12 m
unit variance, have a multivariate normal distribution. Of interest
is the probability that exactly r of these m variables are positive.
If the variables are mutually independent, the problem has the
binomial solution However, when the variables are dependent,
no simple solution exists. In this paper we shall consider the case
when the variables have common correlation p, < p < 1. P (p)
m-1 r:m
will denote the probability that exactly r of the m variables are
positive. That is,
Pr:m(p) = E P(Xil > Xir > ; Xir+1 < -*-Xim < 0)
where the summation is over all partitions fi .... ; i ,... ,i ]
1 1 r r+1 mJ
of the set {l,2,...,m}. Since X .X^** >xra are identically distrib
uted, the above equation may be written as
Pr:() = (r)P(Xl > 0 Xr > 0; Xr+1 < 0 X. *=
0).
Letting g(x ,x x ) represent the density on X ,X
1 Kl 1 2i
,X
m
we have


r
1
1
?
3
4
5
6
7
6
9
10
11
12
13
14
15
16
17
1
2
3
4
5
6
7
0
9
10
11
12
13
14
15
16
17
13
19
20
21
97
APPENDIX 4 (Continued)
n
15
16
17
18
0 .OCGC
0 C C C 0
C.CCCC
O.OCCO
.CCGC
c. c c c c
c.cccc
0.0 C C 0
0.C0C1
c.cccc
c .ccoo
o.ccco
O.C935
C C C 11
C.CC03
0.0001
0.0383
C.C158
0.CC6C
0.C022
0.1899
0.1CC8
C.C49C
0.0220
0.5CCC
fj O 1 C.C
C.2C55
0.1152
0.8101
0.6641
C.5CCC
0.3452
0.9617
0.8592
C.7945
0.6548
0.9965
C.9842
C.551C
C. 8848
0.9999
0.9989
0.9 9 4 C
0.9780
1 .COQC
l.CCCG
C.9997
0.9978
1.CC0C
l.CCCC
l.CCCC
0.9999
1.CC0C
l.CCCC
l.CCCC
l.CCCG
l.CCCC
l.CCCC
l.CCCC
l.CCOC
1.0000
l.CCCC
19
2C
21
22
o.cccc
C.CCCC
C.CCOC
O.OCCO
o.cooc
C.CCCC
c.ccoc
O.OCCO
o.cccc
C.CCCC
c.cccc
o.ccco
o.cccc
O.CCCO
c.cccc
O.OCCO
0.CCC7
C.CCC2
C.OCOl
C.OCOl
0 .CC92
C.0C36
C.CC13
0.0GC5
0.0595
0 C 2 6 6
C.C128
0.0C54
0.219C
C.1282
C.C697
C.0353
0.5C0C
C.353C
C.2308
0.1401
0.7310
C.647C
C.5C0C
0.3598
0.94C5
C.8718
C.7692
0.6402
0.9908
0.9714
C.53C3
0.8599
0.9993
0.9964
C.9872
0.9647
l.CCOC
0.9598
C.9987
0.9946
l.COCC
l.CCCC
0.5589
0.9995
l.CCOC
l.CCCC
l.CCCC
0.9959
l.CCCC
l.CCCC
l.CCCO
l.OCOO
1 .cooc
l.CCCC
l.CCOC
l.CCCO
l.CCCC
l.CCCC
l.OCOO
l.CCCC
l.OCCO
1.0000


APPENDIX 4
P (Y=r)
n
r
'A
4
5
6
1
0.5CCC
0.1755
C.0489
0.0115
2
l.CCCC
0.3245
C.5CCC
0.2348
3
l.CCCC
C.951
0.7652
4
l.CCCC
C.98 85
5
1.00C0
7
£
9
10
1
0.C024
C.CCC4
C.CCC1
C.OlCO
2
0.0904
0.0298
C.0087
0.0023
3
0.5CCC
C.2695
C. 1234
0.0493
4
0.9096
C.71C5
C.5CCC
0.2934
5
0.9976
0.97C2
C 8 76 6
C.7C66
6
1 .CCCC
0.9996
C.9913
C.95C7
7
l.CCCC
C.9999
0.9977
8
l.CCCC
L.CCCC
9
l.OCCO
11
12
13
14
1
O.CCCC
C.CCCC
c.ccoc
o.occo
2
C.C0C5
C.CCC1
C.CCCC
O.OQCC
3
0.0175
0.0C57
C.CC17
O.OOC5
4
0.1499
C .0678
C.C277
0.0103
5
0.5C0C
C.3111
C 1717
0.0351
6
C. 8 50 I
0.6869
C.5CCC
0.3246
7
C.9825
0.9322
C.8283
0.6752
6
0.9995
0.9943
C. 9723
0.9149
9
l.CCCC
C.9999
C .9983
0.9397
1C
l.CCCC
l.CCCC
l.CCCC
0.9995
11
12
13
l.CCCC
l.CCOC
l.CCCC
1.0000
1.0000
l.GOCC
96


72
It is unlikely, from the expressions given for and S^, that
a simple general expression for the kth term exists. Although addi
tional terms in the series could be determined, it was seen from numer
ical examples that the series converges quite slowly, especially for
large values of rho. Some of these results are given in Table 7.
TABLE 7
ERROR INVOVED IN COMPUTING P (0) WHEN THE SERIES
m '
IN RHO IS TRUNCATED AFTER FIVE TERMS
m = 5
ifc 10
Absolute
s,
Absolute
S^
P
Error
4
p
Error
4
-1/10
.00001
.00006
-1/15
.00006
.00002
-1/5
.00008
.00101
-1/10
.00029
.00008
1/5
.00048
.00101
1/5
.00318
.00127
1/10
.00003
.00006
1/10
.00022
.00008
It should be noted that this method of computing P (p) is valid for
m
negative values of rho. In fact, if the results in Table 7 are any
indication of the general behavior of the series, we would expect the
fastest convergence for p <0.


CHAPTER 3
NUMERICAL RESULTS
3.1 Exact Results
In general the value of the integral J (p) can only be
K
approximated so that exact results for pr.m(p) are not available.
However, exact values of J (p) and J(p) can be found. Then, since
o 2
J (p ) is independent of m and r, P (p ) r = 0,1,2, and P (p) ,
K r i r i o
r = 0,1,2,3, can be determined.
Since L (t) = 1, we have from Lemma 2, page 25,
o
JQ(P) = 2|h|
r Lo[h(u)¡[
-Mu2
*u2l' h' _1
J
dG(u)
r hr 2
- e du
*/2ri
= 1.
J (p) can be determined indirectly by first finding P (p)
£i
Letting
m = r = 2 in Lemma 2,
P_ _(p) = P.(p) c (2,2) b (2) J (p)
£\ ^ r\ r\
+ c2(2,2) b2(2) J2(p),
so that
P (p) c (2,2) b (2)
<3-1-1> ) l..,t2)
26


78
O
O o
(2tt) 2
a. = -r2jVv [j 2) (j-3)a _
J,1 j(j-1) j-2,1-2
+ (2ij 5i + j -3)a
j~2,i
where a. equals
j, i
tion, the values of
in Table 8.
+ (i+1)(+2)q^_2 i_2], i < j-2, j =
zero if either i > j-1 or i < 0.
Ct-, 3 = 0,2,...,22, were calculated.
J
2,4,...,
Using this rela
They appear
TABLE 8
VALUES OF a., 3 = 0,2,...,22
3
j
a.
3
j
a
j
3
Of.
J
0
.3989422803
8
-1.958451122
16
-105.6166131
2
-1.253314138
10
-4.703578753
18
-326.3330223
4
-.6562337483
12
-12.48581643
20
-1037.319292
6
-.9620889240
14
-35.44811307
22
-3373.253924
Substituting G(§) = G(0) = into the recurrence relation
(5.3.3) for y (§,m) and denoting y (0,m) by y (m), we have
m+k+1
l\k k ~1
21 2 Yk-l(m)J '
f
\2
Since we need y (m) only for even values of k, we can eliminate
K
y, (m) in the above relation. It follows that
'k-l


34
Figure 3.
The function h2
p = 9/10.
Lk[2G(u) 1]
h-1


BIOGRAPHICAL SKETCH
Thomas Ray Hoffman was born January 31, 1945, in Mechanicsburg,
Pennsylvania. After graduating from Cumberland Valley High School
in 1962, he entered Bucknell University where, in 1966, he received
a Bachelor of Science degree with a major in mathematics.
With plans of becoming a high school mathematics .teacher, he
continued his education at Bucknell University as a graduate teaching
assistant in the Department of Mathematics. It was during this year
of graduate study that he was first introduced to statistics. With
his vocational plans shifting towards teaching at the college level,
he entered the University of Florida's graduate program in statistics
where he was employed as a teaching assistant. In 1969, he received a
Master of Statistics degree. From 1969 until the present, he has been
working towards the degree of Doctor of Philosophy with a major in
statistics.
Thomas Ray Hoffman received the Kappa Phi Kappa fraternity
prize when he graduated from Bucknell University. He is a member of
Pi Mu Epsilon mathematics honorary, the Institute of Mathematical
Statistics, and the American Statistical Association.
100


9
The proof of (1.3.1) for p <0 consists of showing that the
right-hand sides of equations (1.2.8) and (1.3.1) are identical.
First we note that
1 Gtcpy) = 1 e^ 7 f e g(t)dt
_ 00
i 2 2 00
= 1 y f e-'-'W g(t)dt
CO
92y2 p" -ipy .....
+ e J e g(t)at
, 2 2 00
y r --i-tcpy .....
= e1 J e g(t)dt.
o
Then using (1.3.2) and writing the right-hand side of equation (1.3.1),
say R, as a multiple integral, we have
00
R (J) J* [l-G(.cpy)]r[GOcpy)]m"r g(y)dy
- (?) J
r r
j-j
t. >0
- J
j < r
r 2 2
y -ispy E t.
j=l J
g(t )...g(t )dt .dt
1 r l r
J-j
t. <0
- J
j >r
m
m-r 2 2 -Jspy E t.
y i=r+l J
e e J g(t )...g(t )dt ...dt
r+1 m r+1 m
' g(y)dy
m
m 2 2
f*\ r f 7
- \r)
>j
00
e
L[r,m]
-9y 2 t
j=i
g(t )...g(t )dt ...dt
1 m 1 m
g(y)dy.


42
KAf-ArAr+l- "O]}
-C) k* >-(?> i> Or) w
m-r
() s (-dj (m:r )p_.,(p)
j=0
J / r+j
3.4 Accuracy of the Results
The integrals J (p ) the constants d (m) and the probabilities
.K K
P (p) were evaluated with double-precision accuracy using an IBM model
m
360 computer. The computed values of J^ip) were checked against the
exact values for k=0 and k=2 and for p = For k = 0 and k=2,
the results were accurate to at least seven significant digits for
fp | < i and to five significant digits for |p | = ^ J, (x-) was
computed accurately to six significant digits for k< 14 and to five
significant digits for 16< k < 22. Hence, for fp|< i we would
expect J^(p), k>2, to be accurate to at least the fifth significant
digit. The computed values of J (p) k< 22, are given in Appendix 1
K
for p = i p = 2(1)25, and for p = , p = 3(1)26.
P P
As expected, with p > i, accurate values of J (p) were not
2 K
obtained using the method of quadrature described in Section 3.2.
Further investigation has to be made in order to find a means of
evaluating J (p) accurately for p > i .


k
O
2
4
6
8
10
12
14
16
18
20
22
80
TABLE 9
VALUES OF y (m), 6 (1/3), S (1/4),
k = 0,2,...,22, m = 10
Yk(10)
0, (1/3)
k
0.(1/4)
k
.0909090909
.3989422803
.1591549430
.0163170163
-1.253314138
-1.000000000
.0032092907
-.6562337483
1.047197553
.0006629400
-.9620889240
.8772981706
.0001413557
-1.958451122
1.279624120
.0000308241
-4.703578753
2.418906535
.0000068352
-12.48581643
5.323901882
.0000015356
-35.44811307
12.95550088
.0000003486
-105.6166131
33.85865868
.0000000798
-326.3330223
93.33839437
.0000000184
-1037.319292
268.1907545
.0000000043
-3373.253924
796.5361116


70
p2 2! (mN¡ /'m-2\ 1 p p 6 6 6 5
s2 2T TTTT \2/ V 2 J 2 " 'J ?x7?x7'olT" Fi~
0 0 1 2 3 4
g(x x ; 0) dx . dx
1 m 1 m
p2 m(4> r V>-]4 m-4
2 4 L V2''
m m
(4)
Although one might hope for a general expression for S such
, K
hopes diminish after evaluating the next two terms in the series.
With k=3, we must consider assigning either one three" or three
ones" to the N k.'s. (The choice of one "two" and one "one" results
J
in an even powered derivative.) The assignment of the three "ones"
can result in two types of products with only odd exponents on the 6's.
The first type, 6^6i25364> can be formed bY ) [2(m-2)](m-3)~V
different assignments, and the second type, 6j_ 6j_ 6j_ can be
1 ^ o 4 i) b
. j (m-2Vm-4\ 1 . . ..
formed by \^J 2 jy 2 ^7- different assignments of the three ones .
Consequently,
3
, P_
3 3!
f
(m)
CO
f
CO
r
l3:
\2 J
J '
O
*J
0
g 3 g3
rr"^T gU x_;0)dx1...dx
JX1 0X2
m
m
+ lirhr (2) [2(m-2)](n,-3) /-J*
3D CO *3
6J 6 6 6
6x 5x 5x 5x
0 0 1 2 3 4
g(x x ;0)dx . dx
1 m 1 m


79
1 ¡ m k(k-l)
"^k m (m+k+1) (m+k) [_2k + 4 ^
k-2
1
(m)J k = 2,4,
This relation can be easily programmed to evaluate y (m), k = 2,4,.
K
for a given value of m.
Numerical examples were used to investigate how quickly the
series,
(5.3.4)
h-1
i 00
P (p) = (2n) 2 Yfi E 3 (p) y (m) ,
m
k=0
converges. As examples, we include the numerical computations for
m
= 10, p =1/3, 1/4. The values of y (m) 3 (1/3) and 0 (1/4) for
k rk'-' "k
k = 0,2,... ,22 are given in Table 9. In Table 10 is listed
h-1
2 n
(2tt) he E 3 (p) y (m)
k=0 k k
for n = 0,2,... ,22 and p = 1/3, 1/4. The exact value of Pm(p) is
given after n= co As can be seen, absolute errors for p = 1/3 and
p = 1/4 are .00258 and .00122, respectively. In addition to the slow
convergence of the series P^(p ) this method of evaluating P^p ) has
two other disadvantages. First, the expression for P (p) given in
m
(5.3.4) is only valid for p = ^ = 1,2,... Also, the constants
0, (p) are difficult to evaluate for small values of rho.


CHAPTER 2
AN EXPRESSION FOR P (p) INVOLVING TCHEBYCHEFF-HERMITE
r: m
AND LEGENDRE POLYNOMIALS
2.1 Definitions and Properties
Let ck(r,m) denote the kth order Tchebycheff-Hermite polynomial
orthogonal on r = 0,1,...,m. Then c^Cr.m) can be written (see, for
example, Plackett, Sec. 6.5), as
(2. 1.1)
ck(r,m)
(k! )
(2k)!
k
Z (-1)J
j=o
( r \ /^k-jVm-k+jN
Vk-j /Ik A j )
and satisfies the following three properties:
m
(2.1.2) Z c (r,m) =0, k = 1,2,...,
- K
r=0
m
(2.1.3) Z c.(r,m)c (r,m) =0, j / k,
r=0 J k
(2.1.4)
m
ck r=0
m+k+l\
2k+l )
Also, let L^(t) represent the kth order Legendre polynomial
in t. L^it) is given by (see, for example, Abraraowitz and Stegun,
k 2 ^
Chap. 22) the coefficient of s in the expansion of (l-2ts+s ) and
can be computed from the recurrence relation
14


8
respectively. Hence, writing P (p) conditional on Y = y and inte-
r: m o
grating over y, we have
P
r: m
(P)
P (Y > 0Y ,.
1 o
Yr>
9Y
Y < GY ,
r+1 o
...,Y < 6Y |y =y) g(y)dy.
in o' o
Finally, using the independence and identically distributed properties
of Y,Y_....,Y we have
12 m
00
(1.3.1) P (P) = (m) f [1-G(By)]r(G(Gy)]m_r g(y)dy.
r: m \r / c
CO
Results similar to (1.3.1) have been given by Ruben (1954), Dunnett and
Sobel (1955), Moran (1956), and Stuart (1958).
Although the expression for P (p) given in (1.3.1) was derived,
r: m
assuming p >0, Steck and Owen (1962) have shown that it also holds for
p <0 by defining G(6y) in the complex plane. For p <0, 6 is an
imaginary number and can be written 2.0 where
cp =
Then G(6y) equals G(-9y) and is defined by integrating along a path
in the complex plane parallel to the x-axis from -00 + -cpy to + icpy.
That is,
(1.3.2)
G(^cpy) = e
2 2
-lixpy
g(t)dt.
00


47
The standard deviation has been computed for n< 50 and is given
in Table 1.
4.2 The Null Distribution
Clearly the test statistic, Y, has a discrete distribution,
taking on values 1,2,...,n-l with positive probabilities. The proba
bility of the event, [Y= r], r = l,2,...,n-l, can be written as
Pn(Y=r) = I P(CPil = 1 ^ir= 1^ir+1 = 0 cPin = 0> *
where the summation is over all partitions fi ,...,i ; i ^,....i }
w 1 r r+1 nJ
of the set [1,2 ,. . ,n]. Then, since the variables cp ,cp . ,cp are
1 z n
identically distributed,
pn(Y=r) = (¡!) P^1 = 1

n \r / 1 r r+1 n
= (n) P(Y Y>0, . ,Y Y > 0 ; Y -Y<0,
\r/ 1 r r+1
. . ,Y Y < 0) .
n
After the transformation
U. = (Y. -Y)
J J
(
\n-l
J
we have
P (Y= r) 5 P(U >0 U >0; U <0,. . ,U <0) ,
n \r/ 1 r r+1 n
where the normal variables U ,U ,...,U each have mean zero and
1 II
common variance and correlation given by


82
5.4 Using Moments of Extreme Order Statistics
If we let f(u;m) represent the density on the largest order
statistic in a sample of size m from a normal distribution, then
d . m
f(u;m) = G(u)
.m-1 .
= mG(u) g(u),
-CO < u < 00.
Using the representation of P (p) given in (5.3.1), we can write
m
h-1
(2tt) 2 h? p h-1
Pm(p) = J g(u) f(u;nvfl) du.
m +1
The integral, say I, above can be simplified by successively integrat
ing by parts. For example, after integrating by parts twice, we have
I = / g(u)h 1 dG(u)
im-1
CO
r .h-1 m-1
+ J (h-1)u g(u) G(u)
du
-00 _co
n h-1 p .h-2 i
= 0 + J u g(u) dG(u)
(h-1)(h-2)
m
>(h-2) p 2 .h-2 un-2 ,
J (u -1) g (u) G(u) du
(h-1)(h-2)
(m+2)
gif f *- 8<)h-3 -K-)
. m+3
It can be shown by induction that after integrating by parts k times,


90
APPENDIX 2 (Continued)
m
k
13
19
2C
21
0
C.C5263156
C.05CCCCCC
0.04761905
C.C4545455
2
0.19172932
C.185C64S4
0.1738C670
C.17292490
4
Q. 16366936
C. 164 14 45 5
G. 16414455
0.16377318
6
C.07172G4 1
C .0766CC79
0.081C5504
C.08511199
Q
<->
C.C1762539
0.02C67C05
Q.C2375863
C.02686559
1C
C.C0241458
O.CC322t39
G.G0416566
C.0C521391
12
Q.CC017309
C.CCC27695
0.CCC42265
C.GO060912
14
0.CCOO0570
C.CCCG1211
C.00C02307
C.0CCC4C38
16
C.CC00CCC7
C.CCCC0C22
0.0CCCCC59
C.C C0CC137
18
0. CCOCCCQC
c.cccccccc
0.CCCCCC01
C.0C0G0C02
20
0.CCCCCCG1
C.0CCC0CG2
22
23
24
25
Q
0.C4347326
C .04 166667
0.C4CCCCGC
C.03846154
2
0.1673913C
G. 16 2 17545
G. 15726466
C. 15262515
4
0. 16309922
0.16 217549
0.161C6101
0.15978275
6
0.0837806 1
C .06208612
0.C55C5671
C.0977166C
3
0.02996C7C
0.03 3G15 15
0.C3602C93
0.03894997
1C
j.CO637367
C.GC763419
0.CC857407
C.01038666
12
0.0003 42 3 2
C.OC112522
0.CC145975
C.0C164685
14
G.CC0r'6602
C.CCC10212
0.CCC15C82
C.CC021423
16
O.CCOOC284
0.OC COG 536
0.CCCCC642
C.0CC01557
le
C.COQ 1030 6
C.CCCCCC15
0.CCCCCC33
C.CCGC0G66
2C
C.C C000300
G.OOCCOLCC
G.CCCCCC01
C.0CCCGG01
22
C.GCO-COCC
C CLuCOCCC
O.CCCCGCOC
C.OCCCCOOO
24
o.eccocccc
C.CCCCOCCO


CHAPTER 1
INTRODUCTION
1.1 Introduction
In a recent paper, David (1962) suggested using the number of
observations larger than the sample mean as a test for the homogeneity
of a random sample. Assuming a normal population, he showed that the
proportion of observations larger than the sample mean has an asymptotic
normal distribution. However, David did not discuss the small sample
distribution for the test statistic.
The work on this dissertation began in search for the small
sample distribution for David's statistic. How'ever, this work soon
led to the more general problem of finding the probability that part
of a set of equicorrelated normal variables are positive and, in
particular, the problem of evaluating a multivariate normal integral
expression for the probability that all the variables are positive.
Much research related to the multivariate normal integral has
been published. Gupta (1963b), in addition to an excellent survey
paper, gives a complete bibliography of articles related to the multi
variate normal integral. However, only a few of these articles offer
a practical method of evaluating the integral. Also, although Steck
(1962) gives a relation connecting the results for positive and nega
tive correlation, no direct method of evaluating the integral has
1


25
Lemma 2
P ,m(P)
r: m
m
S
k=0
k even
ck(r,m) bk(m) JR(P)
where
/i*'3 k
(k! )
<2k): J=o
/ r ^ f2k-j\ (m-k+j \
cu(r,m) /01^t 2 t 1) (^.jy \ k ) \ j J
b (m)
k
m! (2k+l)!
(k!(m+k+1)!
and for even k, Jk(p) is "the integral
i 1
J (p) = 2 ¡h |£ f L [h(u) ]
r,-iu2i|h|-1
dG(u)
with
P =
h+1
and
2G(u) 1
h(u) = 4
21 r* i.2
h > 0
u
r 4t-
^r e* dt h < -m, m ^ 2 .
^V2tt o
L, (t) is defined by the recurrence relation
k
L (t) = 1
o
L (t) = t
2k1 k1
V = \-X (t) Lk-2(t) k 22


BIBLIOGRAPHY
Abramowitz, M. and Stegun, A. S., eds. (1964). Handbook of Mathemat-
ical Functions. National Bureau of Standards, Washington.
Cramer, H. (1946). Mathematical Methods of Statistics. Princeton
University Press, Princeton.
David, H. T. (1962). The Sample Mean Among the Moderate Order Sta
tistics. Ann. Math. Statist., 33, 1160-1166.-
Dunnett, C. W. and Sobel, M. (1955). Approximations to the Probabil
ity Integral and Certain Percentage Points of a Multivariate
Analogue of Student's t-distribution. Biometrika,42, 258-260.
Flora, R. E. (1965). The Problem of Classifying Members of a Popula-
tion into Groups. Ph.D. Dissertation. Virginia Polytechnic
Institute.
Gupta, S. S. (1963a). Probability Integrals of Multivariate Normal
and Multivariate t. Ann. Math. Statist., 34, 792-828.
Gxrota, S. S. (1963b). Bibliography on the Multivariate Normal Inte
grals and Related Topics. Ann. Math. Statist., 34, 829-838.
Kelley, T. L. (1948). The Kelley Statistical Tables. Harvard
University Press, Cambridge.
Moran, P. A. P. (1956). The Numerical Evaluation of a Class of
Integrals. Proc. Cambridge Philos. Soc., 52, 230-233.
Plackett, R. L. (1960). Principles of Regression Analysis.
The Clarendon Press, Oxford.
Ruben, H. (1954). On the Moments of Order Statistics in Samples from
Normal Populations. Biometrika, 41, 200-227.
Saw, J. G. (1958). Moments of Sample Moments of Censored Samples
from a Normal Population. Biometrika, 45, 211-221.
Saw, J. G. and Chow, B. (1966). The Curve Through the Expected
Values of Ordered Variates and the Sum of Squares of Normal
Scores. Biometrika, 53, 252-255.
98


75
P (0)
m
o= i
O 1. /% 1c TT1
(2tt) h£ E 8 (§;p) J [G(u)-G(§>] G(u) dG(u)
k=0 k o
Next, defining y (§,m) by
&
y. (§,m) = f [G(u) -G(§)]k G(u)m dG(u) ,
we can write
h-1
- CO
p (p) = (2rr) h7 E 8 (S;p) y. (§,m)
m
k=0
A recurrence relation for y (§,m) can be found by first writing the
integrand as
G(u)m[G(u) G(§) ]k = G(u)m+1 [ G(u) -G(§)]k 1
- G(§) G(u)m [G(u) G(§)]k_1
and then integrating by parts. We have
y.(S,m) f GCu)1^1 [G(u) -G(§)]K_1 dG(u)
k-1
- G(§) / G(u)1" [G(u) -G(§)]k 1 dG(u)
= i G(u)m+1 [G(u) G(§) ]k
- G(u)m [G(u) -G(?)]k dG(u)
o
- G(§) Yk_1(§-m)
k m_+- "
= k [1 G(§)] yk(§,m)-G(§) y^iS.m) .


55
Similarly.
3JV
3^V l^V + ^l^V
4^V 4^ + n 1% + 2 4i,J'V 3^V ~ 61^V 2^V
n
6 'i 4
n 2^V J1^V
Finally, the constants, C.(p ,k), can be determined as functions of
J
H.(p ,k) by substituting the power series representation of ll and
J ^ k ^
equating coefficients of n(j). With k = 2, we have
c- 2 1
2SV 2^V 1% n
00 p 00 |
Z H (p ,2) n(j) -j Z H.(p ,1) n(j)
j=0 J Lj=0 J ^
2
. i ]
n+2 L 2 J
n+2
£ H.(p ,2) n(j) Z H.(p rl) n(2j)
j=0 J j=0 3
- 2 Z ZH (pv,l) H (pv,l) n(j+k) Z 2J n(j+l)
j Z C.(PV,2) n(j) .
j-0 J
Equating coefficients of n(j), we find that
Co(V2) W2) H0(^v1}
C1(PV,2) = H1(pv,2) Ho(pv,l) H1(pv>l) 1,
C2(PV2) = H2(PV2) VV^ 2Ho(PV>:L) H2(pv1) 2>


21
an even or odd function in t, depending on whether k is even or odd,
respectively. Also, when rho is positive, G(Sy) is the normal distri
bution function which implies l-2G(0y) is an odd function in y.
Therefore, L [l-2G(0y)] is an odd function in y when k is odd and an
K
even function in y when k is even. Since g(y) the normal density
function, is an even function in y, it follows that J (P), rho posi-
K
tive, equals zero when k is odd, and for even k
00
Jk(p) = 2 f Lk[2G(0y)-l] g(y) dy.
o
After making the transformation u = 0y, J, (p) becomes
K
00 1 U
2 r i q2
Jk(p) = 0 J e du .
v2TT
o
Finally, letting h = and writing the integration with respect to
0
the distribution function G(u), we have
Jk(p)
= 2h
1
r
r-^u -ih-l
j Lk[2G(u)-l] |^e J dG(u) ,
i
0 ^
where p = ^ and 0 < p < 1 imply that h > 0.
1+ 0
Next consider the integral J (p) when rho is negative. Now
K
0 = £ is imaginary and is written as 0 = Jjp where cp =
(1-0)
(1-p)'
Hence, the function G(0y) appearing in J (p) is complex and in order
K.


TABLE 10
THE FIRST n TERMS IN THE SERIES P (p),
m
m = 10, p = 1/3, 1/4, n = 0,2,... ,22
n
p = 1/3^
p = 1/4*
0
.128564
.157459
2
.056070
-.020116
4
.048604
.016459
6
.046344
.022788
8
.045362
.024757
10
.044848
.025568
12
.044546
.025964
14
.044353
.026181
16
.044222
.026309
18
.044130
.026390
20
.044062
.026444
22
.044011
.026481
CO
.043753
.026603
"^Tabulated
1
entries are (2tt)s
1 n
22 Z 0 (l/3)y (10).
k=0
* A n
Tabulated entries are (2rr) E B, (l/4)y, (10).
, k Tk


46
The covariance of Y. -Y and Y -Y and variance of Y -Y are
j k J
cov (Y J, ) cov (Y Y) cov (Y,Y,)+cov (Y, Y)
J k J k
111
n n n
1
n
and
Var (Y 3 2 cov(Y.,Y) + Var (Y)
J J
i -* + i
n n
1 -
respectively. Hence,
P =
1
n
1 i
n
-1
n 1
Therefore,
cov Pj ,9k) P2 (- ^l) V
1 "I
sin
2rr
1
n-lj
and
n
Var (Y) = Var ( S Cp. j
Kj = l 3
n
= £ Var (cp.) + 2 £ £ cov (9. ,cp )
j=l J j < k J
n n(n-l) .-If 1
" 4 + 2tt S1R V n-1


33
x
.2
Figure 2. The function h L^[2G(u) -1]
p = 1/2. '
2i h-1


30
m
in
m
E P (p)
= E
E
r:m
r=0
r=0
m
k=0
= E
r=0
m
b (;
o
II II
II M
o
1
m+1
k k
r:m
From Lemma 1,
P (p) can also be computed exactly in the case when p =
r: m
Pr:n
= (r
f [1-
\r /
CO
since 9=1 when
1
2 '
But
C)
G(y)]r
[G(y)]]
= x {[G(y)1""r [1-G(y)]r g(y)l
where the term inside the braces is the density on the (m-r+l)st
order statistic from a normal random sample of size (m+1).
Therefore,
P () = 1-
r:mv 2J m+1
r = 0,1,...,m.
,1,
This last result implies that J () = 0, k^2. From Lemma 2,
K. a
m
Pr:m(l) = E bk(m) Ck(r'm) Jk(l>
k=0
k even
1 m 1
- + E b (m) c (r ,m) J (-).
m+-l k k k 2
K £
k even


THE PROBABILITY THAT PART OF A SET OF EQUICORRELATED
NORMAL VARIABLES ARE POSITIVE
BY
THOMAS RAY HOFFMAN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1972


64
Differentiating with respect to rho, we have
5
Â¥
m
_ i S t.x,
00 00 j j
T i=l J J
J e s(xi V p)dxr
dx
00 Oj
m
= E E t t, e
. -. i j
- ^ E t2 p E E t.t
2 J 3 k
Since, for < p <1, g(x ,x ; p) is a continuous function
ra-1 1 m
5
in p, t g(x ,...x ; p) exists and is integrable. Therefore, differ-
op 1 m
entiation inside the integral is permitted, and, at the point p =0,
we have
m
00 CO
(5.2.4)
L Z t .x.
3 3
r r i 1 j=iJ J
L j_ Lsp- S dx
m
fP=0
= E Z t.t. e
i 1 m 2
_ £ r
2 ,
J=1 J
Next, we consider the characteristic function of X,...,X when p = 0.
1 m
From (5.2.3) we have
m
00 i. Z t x
j=l J J ,
e g(x
CO
. x ; 0)dx ...dx
m 1 m
= e
.1" t*
2 A J


TABLE 2
THE CUMULATIVE DI
STRIBUTTON
OF Y, n= 19
Approximations
Small
Sample
Edgeworth's'
r
Distribution
Normal
Expansion
i
.0000
.0000
.0000
2
.0000
.0000
.0000
3
.0000
.0000
.0002
4
.0000
.0000
.0007
5
.0007
.0011
.0029
6
.0092
.0112
.0142
7
.0595
.0639
.0638
8
.2190
.2231
.2175
9
.5000
.5000
. 5000
10
.7810
.7769
.7825
11
.9405
.9361
.9362
12
.9908
.9889
.9858
13
.9993
. 9989
. 9971
14
1.0000
1.0000
.9993
15
1.0000
1.0000
.9998
16
1.0000
1.0000
1.0000
17
1.0000
1.0000
1. 0000
18
1.0000
1.0000
1.0000
+
Uses
first two moments
of Y.
+
i _
Uses first four moments of Y, -Y.
(n-r)


28
Finally, making the polar transformation,
u^ = r cos 8
we have
u = r sin 6,
Ci
P2(P)
00 r
-/ {
cos 1(p)
2tt
}
d6 rre
S- r
dr
1 I, V
= -cs (-p)
1 /n -1 .
= 2tt \2^ + Sin P)
11 -1
= + sxn p .
Next we need the values of c, (r,m) and b, (m) for
k k
k = 2. For k = 0, c (r,m) = 1 and for k = 2,
o
c2(r,m) = r(r-l) r(m-l) + ^ m(m-l).
12 1
Thus for m = 2, c (r,m) equals -z, , and for r = 0,1,
C 3 O
respectively, and for m = 3, cr>(r,m) equals 1, -1, -1, and
r = 0,1,2, and 3, respectively. The constants b (m) and br
o
given by
b (ra) = -i- ,
o m+1
, 30
2(m) (m+1) (m-i-2) (m+3)
Substituting m = 2 and m = 3 gives
b (2) = i ,
o o
bo(3) = 4
V2) 2 -
b2(3) = 1
k = 0 and
and 2,
1 for
(m) are


6
1.3 A Transformation Simplifying P (p)
r: m
Consider the m variables X defined by
1 2 m
X = Y 3Y
11c
X2 = Y2 9Yo
X = Y 6Y ,
mm o
where the (m+1) variables, Y ,Y,.. ,Y have a multivariate normal
o 1 m
distribution with mean vector zero and dispersion matrix I, and 8 is
an arbitrary constant. It follows that X^,X ,...,X have a multivari-
J 12m
ate normal distribution with
E(x') = E(Y.) 0E(Y )
11 o
= 0,
Var (X!) = Var (Y.) + 02 Var (Y ) 29 Cov (Y.,Y )
i i o i o
= 1 + 02,
Cov (x',x'.) = Cov (Y.-0Y Y. -0Y )
i j i O J o
= Cov (Y.,Y.) 9 Cov (Y.,Y )
i J i o
- 9 Cov (Y Y.) + 02 Cov (Y ,Y )
o j o o
= 0
Therefore, assuming p > 0, if we define 8 by
0 =
(1-p)
i
2 2-1
9 (1+8 ) equals p and the variables
2 t
(1+0 )
(1+92)2
m
(1.92)4
have the same distribution as X,X,...,X defined in Section (1.2).
12 m
That is,


TABLE OF CONTENTS (Continued)
CHAPTER Page
5 OTHER METHODS OF EXPRESSING P (p) 62
ra
5.1 Introduction 62
5.2 A Power Series in Rho 62
5.3 A Series Resulting from an Inverse
Taylor Series Expansion of g(u) 73
5.4 Using Moments of Extreme Order Statistics .... 82
APPENDIXES 84
BIBLIOGRAPHY 98
BIOGRAPHICAL SKETCH 100
v


11
-1 2
Substituting -p(l-p) for cp ,
in
m
* S t.
R = J (2tt) 2(liD) 1+ (m-1)] ^ e j 1
[r,m]
m 2
P ^ "j)
e2[1+J(m-1)p] dt ...dt .
1 m
Finally, making the transformation
t. =
3
(1-p)
"T t j 1,2 ,, ,in,
we have
m m-1
P. = (jj) f (2tt) 2(1^> 2 [ 1+ (m-l)p ]
[r, m]
ri
m
exp <-
[1+ (m-2)p ] Z x. 2p E Z x.x '
3=1 3 j >dx....dx
1 m
2(l-p)[l+(m-l)p]
and the proof is complete.
The results of this section for positive and negative rhc
be summarized in the following lemma.
Lemma 1
Pr:m 9 =
(1P )
can
where


35
j = O 1,. .. ,200. Then J. (p ) can be approximated by
where
-i u h-1
i oo25 200 r-'?uinJ
J, (p) = 2h* Z c. L (. 005j) | e J j
k J j=o J -1
Co 200 1
odd,
J4 'J
\,2 j even,
and u_. is the value of u satisfying G(u) = .5 + (,0025)j.
The Case When Rho Is Negative
With p <0, the Legendre polynomial has the imaginary argument
Li(u) where f(u) is given by
9 u i +2
f(u) = P e^ dt.
V§tt o
For given values of u, the function f(u) can be evaluated quite rapidly
by first expanding the integrand in a Maclaurin series and integrating
term by term. That is,
u 2j
f(u) = f E t dt
V^TT o j=0 2Jj!
,u t2J
j=0 0 2Jj!
dt
CO
j=0 */2tt
u2J + i
2J(2j+i)j:


45
E(Y)
n
= E E(tp.)
j = l
n
= E P(cp. = 1)
j=l J
n
= Z P(Y. Y > 0)
j=l J
n
2
since Y Y is symmetrically distributed about zero. Before
calculating the variance of Y, we first need the variance of cp^.
covariance of cp and cp We have
J k
and
Var (cp.) = E (cp2) [E (cp .) ]2
J 0 J
= E(=p.) (|)2
1
~ 4
COV (Cp.,cp ) = E(cp.Cp ) E (Cp .) E(Cp )
J K J K J K
= P(Y Y > 0 Y Y > 0) j .
J k 4 .
But the probability above is identical to P (p), where p equals
corr (Y. Y, Y, -Y) =
J k
cov (Y.-Y, Y -Y)
J K
Jw
ar (Y.-Y) Var (Y,-Y)
J k
and


68
(6 5_)
v 1 2'
k9 k1M
(66) 2...(S 6 ) N
1 o m-1 m
where N = ( j and where the summation is over all integer values of
N
k., ,k.,.. . ,k satisfying Z k. = k and k. > 0, j = 1,2,... ,N.
1 z N j-1 3 3
Using this expansion and the value of the integral given by (5.2.8),
we can determine the first few terms in the series expression (5.2.7)
for P (p). Denoting the (k+l)st term in the series by S, for the
m k
first two terms, we have
and
O co
CC CO
1 ,
S1 = PV J--- y k '1! k ^16o) 1(6163) 2--(6 l6 } N
1 1. J J /1 k k 12 1 3 m-1 m
o o 1 N
g(x x ) dx . dx
1 m 1 m
Since
since
S k. = 1, there are N = terms in the sum. Therefore,
j=l J X 7
X ,...,Xm are identically distributed, it follows that


24
1 r^y 1 i-t2 ..
G(^cpy) = + JL J e* dt
O V2TT
2 + -3(CDy) .
Returning to the integral J (p) we can now write
IV
Jk(p) = f Lk[-23(Cpy)] g(y) dy.
Although the Legendre polynomial has an imaginary component, its
definition and recurrence relation still hold. In fact,
L, [-2^,8 (cpy)] is an even function in y when k is even and an odd
function in y when k is odd. Therefore, as in the case when rho is
positive, J (p) equals zero for odd k and for k even
K
J (p) = 2 J L [29(cpy)] g(y) dy.
After making the transformations u = cpy and h = ,
J, (p) becomes
k
9 cp'
Jk(P)
= 2(-h)* f
i
Lkr2X 8(U)] |^e
2i
-tu j
-h-1
dG(u) ,
where p = r-1 and < p < 0 imply that h<-m, m S 2.
h+1 m-1
The following lemma summarizes the results of this chapter.


18
2.2 The Fundamental Result
In this section we use the results of the last section and
Lemma 1, page 11, to obtain a finite series expression for P (p ).
First we let p = 1 G(0y) in equation (2.1.6). Then we have
m
E j[ l-GC 0y) ]x [ GC Gy)]x'ck r=0
ml (k!)
(m-k)!(2k)I k
r L, [ 1-2G( 6y) ]
Multiplication of the above equation by g(y) and integration with
respect to y gives
(2.2.1) J
- r=0
in / v
E ^)[l-G(ey)]r[G(0y)]m-r ck(r,m) g(y)
2 00
mI(k:) r I L [1-2G(0y)] g(y) dy
(m-k)! (2k)! >J_ ^ k
Now taking the left-hand integral inside the summation and defining
Jk(p) by
(2.2.2) J (p) =J L [l-2G(9y)]g(y) dy,
2 2-1
where p = 6 (l+8~) equation (2.2.1) becomes
m 00
/i TTI \ T fYl T*
j[lG(8y)] [G(0y) ] g(y) dy
k=0
m.' (k!)
(m-k)!(2k)! k
r J(p).


2
3
4
5
fe
7

9
10
1 1
12
12
14
15
16
17
13
19
20
21
22
2
3
L
5
fe
7
8
9
10
11
12
1 3
14
15
16
17
16
19
20
2 1
22
93
APPENDIX 3 (Continued)
1/18
0.25888
0.12827
C 0 7 5 9 9
C.04285
0.02475
0.01460
C.0C878
C.OC539
O.OC33
O.OC212
0.00137
C.0CC9C
0.0GC58
0.0CC4C
C.0CCZ7
c.occie
0.000 12
O.OGOC9
C.0CCC6
C.OCC04
C.0CCG2
1/19
C .25828
C 13757
C .07527
C .04222
C G 2 4 2 3
C .G1420
C .00849
C.00 5 16
C C C 2 2 C
C .00201
C .00 128
C .CCC83
C .CCC54
C .C0C36
C C 0 C 2 4
C .CCClfc
C .COCI 1
c .cccca
C .CCCC5
C .CCCC4
C.CCCC3
1/20
C.25796
0.13694
C.C7462
C.C4165
C.G2276
0.C1334
C.CC821
C.0C496
C.CC3C5
C.C019C
0.C012C
C.CCC77
C.CCC5C
C.CCC33
C.CCC22
C.CCC15
C.CCC1C
0.CCCC7
0.CCC05
C.CCC03
C.CCC02
1/21
0.25758
C.13637
0.C7403
C.04113
C.C2335
C.01352
C.C797
C..CQ478
C.00292
C.C0180
C.C0113
C.C0072
0.CCG46
C.C003C
C.CC02C
G.C0013
C.C0CC9
C.G0G06
O.COCC4
C.C00C3
C.C0C02
1/22
1/23
1/24
1/25
0.25724
C.25692
0.25663
0.25637
C. 12586
C. 13 5 38
C. 13495
C.13455
C.0735C
0.0730 2
C.C7253
C.07217
0.04067
C.C4C24
C.C3985
C.03950
C.02297
C.02263
C.02232
0.02203
0.01322
C.01256
C.01273
0.01251
C .0077 5
C G 0 7 5 6
C.CC73e
C.00722
0.00462
C C 0 4 4 8
C.CC435
C.00423
C.0C28C
C .CC27C
C.CC26C
C.C0252
O.OC 172
C .00165
C CC 158
0.C0152
0. OCIO 7
C .00102
C.CCC97
0.C0093
0.0006 6
C.CCC 64
C C C C 6 1
0.C0053
0.0 C C 4 2
C .CCC41
c.ccc3e
G.00036
0.00028
C.CCC 26
C.CCC24
C.CC023
0.00018
C.C0C17
C.CCC16
C.C0015
C.00012
C.CCC11
C.CCC1C
C.C0QC9
0.OCOGE
C .CCCC7
C.CCCC7
C.00006
0.0CCG5
C.CCC G 5
C.CCCC4
C.CQC04
C.0CCC4
C.CCCC3
0.CCC03
G.C0G03
C.OCCC2
C .C0CC2
C.GCC02
C.C00C2
C.0CCC2
C .CCCC2
C.CCC01
G.00001


65
Differentiation with respect to t and t gives
a o
m
00 00
L £ t.x.
r 6 6 j=i 3 3
^T'StT e
_oo -oo a b
g(xr.
.,x ; 0)dx,...dx
m 1 m
so that
- E t^
6 5 2 L Tj
0t3t 6
a b
(5.2.5)
m
00 CO
j i (-\V
E t.x.
j=l J J
CO CO
g(x ,.. ,x 0)dx . .dx
1 m 1 m
= t t e
a b
-i E t2
2 >1 J
Again, differentiation inside the integral was permitted, since the
function x x, g(x....,x ; 0) exists and is integrable. Since
a b 1 m
x
a
*b gU
1 '
; ).
m
after summing both sides of equation (5.2.5) over values of a and b
such that a < b, we have
m
00 CO
(5.2.6)
i- E t .x.
r r 6 6 i -¡=1 3 3
J-L?i^T'5i:g J L x
_oo _co -a dx.
= E E t t, e
a b
a -i Et2
2 J-l J
. dx
m


40
Thus, we must show that the summation above equals one. This summation
can be written as
k
£ (-1)J
j=0
Then, letting
k
£
0= o
0 = k j the result to be proved becomes
Next, we introduce the negative binomial and binomial identities given
by
(3.3.6)
1
(1+a)
k+1
£
r=0
(3.3.7)
Multiplying the left-hand sides of (3.3.6) and (3.3.7), gives
<1"a) = _ =J_ J (.j)1 a
k._ .k+1 k._ k '
a (1+a)
a (1+a) a 0=0
Equating this product to the product of the right-hand sides of (3.3.6)
and (3.3.7), we have
4 i H)'.'. S E C-r (k+/) (*) .r-1
a* 0=0 r=0 0=0 V / W
o k
Finally, equating coefficients of a and dividing by (-1) we have
k
£
0=0
as was to be shown.


TABLE 3
C (p
j PV
2)
X
0
1
2
3
4
5
. 50
Zero
.57079633
.46740110
-.53726893
-3.86976182
-12.0889819
. 55
Zero
.57983932
.49410800
-.48836662
-3.82833586
-12.1396154
.60
Zero
.60792651
.57852616
33227941
-3.70150569
-12.3352921
. 65
Zero
.65821859
.73530568
-.03640528
-3.48374825
-12.8518246
.70
Zero
.73711417
.99616602
.47260438
-3. 17871571
-14.2029896
.75
Zero
.85676747
1.42762451
1.35866689
-2.85548457
-18.0128221
. 80
Zero
1.04137154
2.18241438
3.03241842
-2.93521545
-30.4943719
. 85
Zero
1.34534536
3.67686183
6.76962950
-6.03082075
-84.1011123
. 90
Zero
1.92211072
7.45692711
18.38903662
-35.47326470
-474.5834482
Cn
-a


I
4
(1.2.4)
1+ (m-2)p -p
1
(1-p )[1+(m-l)p]
-P
l-t-(m-2)p
-P
-P
-P
1+ (m-2)p_
Since the determinant of a matrix equals the product of its
latent roots, (1.2.3) can be proven by finding the m latent roots of V.
If X is a latent root of V it must satisfy
(V \I)y = 0 ,
where 0 represents the (mxl) vector of zeros, for at
vector y. Letting X = (1-p) we note that the matrix
only one distinct element. Therefore the rank of (V
there exist (m-1) non-zero vectors y satisfying (V -
(1-p) is a (m-1)-fold latent root of V. To find the
we note that the trace of a matrix equals the sum of
Since the trace of V equals m,
least one non-zero
(V XD contains
- XI) is one and
XI )y = 0. Hence,
last latent root
its latent roots.
X = m (m-1) (1-p )
in
= 1 + (m-l)p,
and (1.2.3) is proven.
-1
To prove (1.2.4) denote V by A = (a ).
op
Then A must satisfy
m
JB -
m
jfj vaj ajS = B '


77
Therefore, it would be better to choose the point G(§) that minimizes
a.(§), j = 0,1,..., especially since the y (§,m)'s are bounded by
J k
(m+1) Since c*j(§) can represented by
a (§) = g(§)
o
Q1(§) = §
a.(S) =
j-2
j:g(§)J i=0
ty £ a .r ,
3-1 j >1
j 2,
where the a. are constants, satisfying
J,i
a. = 0,
J t1
i + j odd,
aj.i<0'
i + j even,
the choice § = 0 clearly minimizes
<2.(§)|. Denoting Q'.(O) by
J J
a., we have
J
(2tt)
Of. = r-r
J j!
Jli
2
J
so that a. vanishes for odd values of j. Since 3 (0;p), say 3,(p ) ,
J K K
is a product of the ct.'s such that Ej. = k, it follows that 8 (p)
Jr k
vanishes for odd values of k, and that the series P (p) contains only
m
even terms.
In order to determine the values for the cr.'s, we first need
J
the coefficients of u1 in the function Qij(u). Denoting this coeffi
cient by or. and using the recurrence relation given by Saw (1958),
J r
we have