THE DISTRIBUTION OF HOTELLING'S
GENERALIZED T2
By
DAVID TIMOTHY HUGHES
A DISSERTATIION 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
1970
Dedicated to my offe Margaret and tom ae'
for their Zove and understanding
ACKNl~OWLEDG EVENTS
I would like to express my deep gratitude to
Dr. J. G. Saw~ for his constant assistance and encouragement.
Without his guidance, this work would not have been possible.
I would also like to thank Drs. P. V. Rao and
Richard L. Scheaffer for their careful proofreading and
helpful criticism. Special thanks must also go to Miss
Gretchen Uzzell for her excellent editing and to Mrs. Carolyn
Lyons w~ho turned a crude manuscript into a masterpiece of
typing.
iii
TABLE OF CONTENTS
Page
\CK'NOWrLEDGEI EN TTS .................... . iii
LIST OF TABLES .......................*** vi
ABSTRACT . ..................... vii
Chapter
I. STATEMENT OF THE PROBLEM .................... 1
1.1 Introduction ................... 1
1.2 Historical Background .......1........ 6
1.3 Su~mm~ry of ResulYts ................... 10
1.4 HoItation, ........................ 11
II. ORTHOSPHERICAL INTEGR~ALS ................... 17
2.1 Introduction ........................ 17
2.2 Some Relationships on the
Orthospherical Integra~s ........... 20
2.3 Evaluation of OrthospherseaZ
IntegraZs .. . . . . . . 28
III. MIOMENTS OF: HOTELLING'S GENERA4LIZED To ...... 44
3.1 In!troduction **********........ 44
3.2 A Method for Evaluating the
3. 3 Th2e Fir~st Two Nom~ents of T ........... 56
3. 4 Evl~zuation~ of Certain Exupectations
3.b5 Su~mmary .. . . . .. .  *
IV. PPOXIMATIONS TO THE DISTRIBUTION OF
HOTELLINlG'S GENERA~LIZED To2 .....~,......... 75
4.2 IntrG~~odu to . . .. . . . 75
.2Avoroximations to th istribto
and Co~nvrison of Percentage
Points of T ........................ 76
4.3 Summary ...,....... ....... ........... 89
TABLE OF CONTENTSContinued
Chapter Page
V. THE POWER OF HOTELLING'S GENERALIZED T 2 .... 92
5.1 Introduction ......................... 92
5.2 An Upper Bound on the Integral
5.3 A Lower Bound on the Integral
of te Poer uncton o To ...... 99
5.4 The Monotonicity of the Power
Function of To2 .................... 110
5.5 Summary .. . . . . . . . 114
BIBLIOGRAPHY ........................................ 117
BIOGRIAPHICAL SKETCH ................................. 120
LIST OF TABLES
Table Page
2.1 Orthospherical Integrals .................... 39
4.1 Upper 5% points of T ........................ 83
4.2 Upper 1% points of T ........................ 84
4.3 Upper 5% points of T ........................ 85
4.4 Upper 1% points of T ........................ 86
4.5 Upper 5% points of T ........................ 87
4.6 Upper 1% points of T ........................ 88
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 DISTRIBUTION OF HOTELLING'S
GENERALIZED T 2
By
David Timothy Hughes
August, 1970
Chairman: Dr. J. G. Saw
Major Department: Statistics
Numerous criteria are available for testing the
multivariate general linear hypothesis. In this work some
properties of one of these criteria, T (a constant times
Hotelling's generalized T 2), are investigated. The T
statistic is the sum of the latent roots of a matrix obtained
from random sampling from a multivariate normal population,
and its distribution depends upon the corresponding popu
lation roots and some degrees of freedom parameters.
The orthospherical integrals involve expectations
of the elements of a matrix having invariant measure on the
orthogonal group. These integrals are defined and a method
for evaluating them is presented. We have derived a special
representation of T which, with the aid of the orthospherical
integrals, enables us to present a method of evaluating the
moments of T. T'he first two nonnull moments are obtained
as an example.
vii
The exact distribution of T in the general case is
difficult to deal with and for this reason, three rather
simple approximations to its distribution are obtained.
All of these approximations are obtained by equating the
moments of T wiith the moments of the approximating random
variable. Accuracy comparisons of percentage points in
certain special cases indicate the usefulness of two of
these approximations.
A result by Wald and Hsu on the power function of
the analysis of variance test is used to obtain an upper
bound on the integral of the power function of T. A
lower bound on the integral of the power function of T
is obtained by using the properties of completely monotone
functions. Completely monotone functions are also used to
demonstrate the monotonicityv of the power function of T.
viii
CHAPTER I
STAiTEMENT OF THE PROBLEMS
1.1 Introduction
Suppose we observe a number of mvariatee column
vectors, X.:1=12..,k j = 1,2,..,n. obtained by
13 1
random sampling from each of k; mvariate normal populations
with common dispersion matrixi, V. Assume that the elemenlts
of the column vectors associated with population i: i=1
2,...,k, have expectations given by the elements of an
mdiensonalmea vecor,9..It is desired to test the
mdiensinal ean ecto, r1
hypothesis
H :ui uz **=u
O r\
against (I.1.1)
H :at least one of the above qualities
a does not hold.
W~ith
N = r n. ( .1 2
i=1
define x. and x by
n.
1
 1
3
and
k ni
x .
i=1 j=1
Construct the i?.x m matrix,
k ni
S2 = C (x..
i=1 j=1 1
 x. ~x..
Y x. ]
(1.1.5)
where (*)' denotes the transpTose of the vector in brackets.
The matrix 52 will be nonsingular with probability one,
whenever S k >m (see [1] or [32]). An appropriate test
statistic for testing (1.1.1) is
k
To = (N k) n. x.
i=1
s Sp1(
 x ] S2~ ( .
x
=(N k) tr S1S21,
where S1 is the m x m matrix defined by
k
S, = n. i.
i=1
 x ]x.
x ] .
The a~ll hypothesis is rejected if
T > t,
where t is chosen to satisfy
(1.1.4)
(1.1.7)
(1.1.8)
P(To2 > tI0) = 2
(1.1.9)
3
for some desired level of significance, a. Thle quantity,
T 2, defined in (1.1.6), has the distribution of Hotelling's
generalized T 2
Mo~re generally, let ~i:i = 1,2,***,vi, be mdimen
sional, normally distributed column v\ectors w~ith mdimen
sional mean vectors, lq, and common m x m dispersion
matrix, V. Suppose S2 iS an m x m central Wishart matrix
(see [36]) distributed independently of Ez,**,[ with
dispersion matrix V and degrees of freedom v2 > i*
Then for testing the hypothesis
H : i ** = p 0,
O 9
against (1.1.10)
Hl :at least one p. 0 ,
a 1
which is the canonical form of the general linear hypothesis,
an appropriate test statistic is Hotelling's generalized
T 2, defined by
To = 2 L q lS2 q~
i=1
=v, tr S1S21, (1.1.11)
where S1 is the mx; m matrix
V1
i=1
It is noted that S1 defined. in 1.12hain general, a
noncentral Wishart distribution with dispersion matrix, V,
degrees of freedom v, and m x m noncentrality matrix
i=1
(1.1.13)
In the first example given in which T w ras defined
by (1.1.6),
(1.1.14)
(1.1.15)
vl = k 1,
V2 = N k,
and
1=~
(1.1.16)
wih e re
(1.1.17)
The general problem just given is invariant under
linear transformations. That is, if instead of observing
gi,***,Lv and S2!, w~e observe zz,***,z~ and S defined by
1 1
(1.1.18)
z. = v. : i = 1,***, v
and
S = AS2A'
(1.1.19)
 5 
for any nonsingulair :n i m natrix, A, the problem remains
the same. The mvariate~ column vet:ors. =;: i = 1, *, v,,
have mean vectors
(i = Ag. : 1 = 1,,v*, ( 1.0
and the hypothesis (1.1.10) is equivalent to
H :51 = 0,
against (1.1.21)
H :at least one 5 0.
a i
This property leads to consideration of test
criteria which are also invariant under linear transfor
mations. Any such~ criterion must be a function of the roots
of the determinantal equation
(S1 aS2 = 0, ( .2
where S1 is defined in (1.1.12). The distribution of such
a criterion is' a function only of m, v v, and the non
zero solutions of
504I' XV = 0. (..3
(At most, min(mn,v,) of AX,***,Xm are noncero.)
A number of test criteria have been proposed for
testing (1.1.10). These include:
6
1. Likelihood ratio: H1+ i ;(1.1.24)
i=1
2. Roy's largest root: max(6 j; (1.1.25)
3.Pillai's criteria: 6 ; (..6
i=1
4C. Hotelling's generalized T :v 0. (.27
i=1
In this paper we investigate some of thie properties of
T = T 2/v2 ..8
for testing the general linear hypothesis. (It has become
standard practice in recent years to study the properties
of T rather thnan T_ itself. We will continue in this
practice and steadfastly hold that the symbol "T" wJill
denote (1.1.28). The symbol "T 2" will be used only when
specifically discussing the properties of (1.1.11) rathier
than (1.1.28). The s?;mibol "tT' should not be confused Uwith
Hotelling's T2 Isee equation (1.2.1)). W~hen discussing
(T 2/v,) we shall use "(T)2" and reserve "T2" for (1.2.1).
1.2 Historical Backgound
In 1931, Hotelling [11] suggested a statistic which
he called T2 as a generalization of "Student's t" ratio.
Hotelling's Tz is in fact a special case of T given in
(1.111)when9, 1.Using (1.1.11), T2 may be defined as
 7 
T2 = vzy'S21~i (.2.1)
The distribution of T2 was derived by Hotelling [1l] under
the null hypothesis, and, in 1938, Hsu [14] derived the
nonnull distribution.
The statistic T (see (1.1.28)] was first suggested
in 1938 by Lawley [26] as a generalization of Fisher's z
(the familiar F ratio) for testing significance in multi
variate analysis of variance. At that time, Lawley obtained
the distribution of
z=10{ my 2 1 To ') (1.2.2)
and showed that the limiting distribution of T as v2 tends
to infinity is the ChiSquare distribution on mvi degrees
of freedom. Hsu [15] also demonstrated this result and
obtained the null distribution of T in integral form. In
this same paper, Hsu derived the first two nonnull moments
of T, but, as w~e shall showz, his second moment is in error.
Somewhat later, Hotelling [~12] suggested T2 as a
generalization of his T2 statistic for measuring multi
variate dispersion. In 1951, the null distribution of T *
was derived by Hotelling [13] for the case m = 2 and for
general v, and v; > 2 in terms of the Gaussian hypergeo
metric function.

Since that time, numerous persons have studied the
distribution of T_ and T. Pillai [29] obtained thie first
three null moments of T aind used these to obtain an approxi
mation to its distribution. Using this approximation, Pillai
and Samson [31] gave upper 54 and 1% points of T for m = 2,3
and selected values of vl and v2. In 1956, Ito [17] ob
tained an asymptotic formula for the null distribution of
T_ as v2 + m and in 1960 extended this result to cover the
nonnull distribution [18]. Bagai [2] derived the null
distribution of T in terms of definite integrals for the
cases m = 2,3. The first tw~o nonnull moments of T 2
given by MIikhail [27] but his second moment, although it
differs from that found by Hsu, is also incorrect.
The distributions of latent roots of random matrices
were expressed by James [20], [22] and Constantine [3] in
terms of zonal polynomials [3], [21]. Constantine [4] used
these results to obtain the distribution of T, in both the
null and nonnull case, in series form. These series con
verge, however, only for IT < 1. In this same paper,
Constantine derived the rth moment of T in terms of the
zonal polynomials for v2 > 2r + m 1 (the moments do not
exist otherwise) and listed the first two null moments
explicitly.
In 1965, Khatri and Pillai [23] derived the first
tw~o moments of T in the linear case (the case of only one
nonzero solution of 1.23 and the same authors used
 9 
these results to approximate the distribution of T in the
linear case [24]. They later generalized this to consider
any number of nonzero roots [25]. Most recently, Davis [6]
showed that, in the null case, the density of T satisfies
an ordinary linear differential equation of degree m of
Fuchsian type and obtains an equivalent first order system.
These results are used to tabulate the upper 5% and 1%
points of To2 1, fOr m = 3,4 [7]. Stein [34] showed that
Hotelling's T2 iS an admissible test of
Ho:p = 0,
against (1.2.3)
Ha:Fp 0 ,
and Ghosh [10] extended this by showing that T 2 iS admis
sible for testing the general linear hypothesis. Das Gupta,
Anderson and Mvudholkar [5] showed that the power of T 2 iS
an increasing function of each X., the remaining X.'s being
held fixed, where Az,**,''m are defined in (1.1.23).
Mikhail [27] and Srivastava [33] later claimed that the
power of To2 depends upon X,,''!,[ only through
i=1
This claim is false, however, since the second moment of
T 2 depends upon
i=1
 10 
Ito [19] gave pow~ier comparisons of To2 and the likelihood
ratio criteria for the linear case and noted little or no
difference in their powers for large V2. For the case
m = 2, extensive power comparisons of T and three other
test criteria for the general linear hypothesis have been
made by Pillai and Jayachlandran [30].
1.3 Summary of Results
Before proceeding with a discussion of the
distribution of T, some preliminary mathematical results
are needed. Therefore, Chapter II is devoted to the
"orthospherical integrals." These integrals are defined
and a method of evaluating them is presented. The ortho
spherical integrals up to order eight are tabulated at the
end of Chapter II. In Chapter III, a general method for
evaluating the moments of T is presented and the first two
nonnull moments are found explicitly. These moments are
used in Chapter IV to obtain three approximations to the
distribution of T. The 5% and 1% points of T are found
for each approximation, and accuracy comparisons are made
with the exact percentage points and Pillai's approximate
percentage points for m = 2. In the case m = 3,4, compari
sons ar~e made w~ith the percentage points given by Davis [7].
Upper and low~er bounds are given for the integral of the
power function of T. In addition we offer an alternative
proof of' the result due to Das Gupta, Anderson and Miudholkar.
 11 
1.4 Notation
Any work in multivariate analysis is greatly
facilitated by frequent use of matrix notation, and it is
also very convenient to abbreviate the distributional
properties of random variables. For these reasons, certain
notational conventions have been adopted. Although many of
these are standard, they will be listed here for reference.
1. Matrices will be denoted by capital letters and
the first time a matrix appears, its rowz and
column dimensions will be given. Thus A:(r x s)
denotes a matrix with r rows and s columns. The
matrix having all its elements zero will be denoted
by (0). "I" will be reserved for the identity
matrix.
2. The elements of a matrix will be denoted by the
corresponding small letter with subscripts to
denote their rowz and column position. Thus a..
denotes the element in the ith row~ and jth column
of the matrix A. The symbol [A)i is equivalent
to a.. and is somretimes used for convenience.
3. An underscored small letter invariably represents
a column vector. Its rowJ dimension will be given
the first time a vector appears. Thus x: [m) denotes
a column vector consisting of m elements. The
12
vector which has all its elements zero will be
denoted by 0.
4. It is sometimes convenient to form row or column
vectors from the rows or columns of a matrix.
The symbol (A).~ will denote the row vector
formed from the ith row of the matrix A. The
symbol (A) denotes the column vector formed
from the jthz column of the matrix A.
5. It is sometimes convenient to partition a square
matrix AZ:(r x r) as
All I Al2
where All is the p x p matrix formed from the
first p rows and columns A; A12 is the p x q
matrix formed from the first p row~s and the last
q columns of A, p + q = r. A21 and A22 are simi
larly defined.
6. The Kronecker product of A: (r x r) and B: (s x s)
will be denoted by A 8 B and is a matrix C:(rs x rs)
w~ith
c~k1)s+i,(1)s+j = akeib .I
7. If x is a random variable having the normal density
with mean y, and variance a this will be denoted
by
13
8. If x is a random variable on (0,=) with densit)
f(x) = {7( ]2(J8/23 V1 xylx{1 21
we shall abbreviate this by
This is to be read as "x has the central Chi
density on v degrees of freedom."
9. If x is a random variable on (0,=) with density
f(xu) = exp ,  {jr( A v]2(u+ 2j)/2 
j=0
x ~jy exp [7),
we shall denote this by
This is to be read as "x has the noncentral Chi
square density on v degrees of freedom with non
centrality paramneter, ."If X = 0, the density;
becomes central.
10. If xt and x? are independently distributed with
xtL ^ X, 2 9)
and
x2 ^ 7~ 22(0),
14
then
z 2 X1
V1 X2
has the noncentral F density on vl and v2 degrees
of freedom and noncentrality parameter, A. This
will be abbreviated
11. If xl and x2 are independently distributed with
X1i ^ X. 2(A
and
x2 ^ X~ 2(0),
then
X2
x1 + x2
has the noncentral Beta density on vl and v2
degrees of freedom with noncentrality parameter,
A. This will be denoted by
u h Se(va~v2,A).
12. If xc is an mdimensional column vector whose
elements have a joint normal density with mean
vector, i: (m) and dispersion matrix V: (m x m),
this will be denoted by
xr N (lcLV).
15
13. Lets x**,xn be independent mvariate column
vectors with
x. lu.,Vi. : i = 1,*** n.
Then wit~h X = (xxl,***,xn and :: = (s**E)
the joint density of the elements of X will be
abbreviated
X mxn(V I)
14. With X an m x n matrix such that
X Nmxn(V I)
the m x m matrix
S = XX'
has a noncentral TWishart distribution with
dispersion matrix V, degrees of freedom n, and
noncentrality matrix 'Ii`I'. This wiill be abbre
viated
S ~ Wm(V,n,?bml').
15. If S is an m x m symmetric matrix whose m(m + 1)/2
mathematically independent elements have the density
f(S) = K (V,v) IS (9m(V2P ex tr V15)
on S positive definite with
Km(V,V) = {2my/2 m"(m1)~YI/4 V v/2 'L
j=1
 16 
this will be abbreviated by
S Wm(Cv,(0) .)
This is to be read as "the m(m + 1)/2 mathemati
cally independent elements of S have the central
Wishiart density with dispersion matrix, V, and
degrees of freedom v."
16. The symbols "a ==> b" and "a <==> b" are to be
read "a implies b" and "a implies, and is implied
by, b," respectively.
17. Suppose it is desired to integrate the function
fCxx,***,xn) over the set
A = {(x ,***,x ] : a. < x.
Then
f(x) d(x)
will be used to abbreviate
bi b
n f ( x t, *,x ). d x * * d x y
ai a
18. For referencing within the text [*] will denote
bibliographical refeyprencs while (*) will denote
references to equations. Thus [8] refers to the
eighth entry in the bibliography, while (4.2.6)
refers to equation six of Section 2 in Chapter IV.
Other special notational conventions will be
introduced as they occur in the text.
CIRAPTER II
ORTHOSPHERICAL INTEGR~ALS
2.1 Introduction
Let R(m x m) denote the group of m x m orthogonal
matrices. Let 4 be an m x m matrix of random variables
with 4 belonging to the group R(m x m), and suppose that 4
has invariant measure on R(m x m). This measure can be
determined in the following manner. Consider any row~ of 4,
say the first row for convenience. Then since 4 is orthogo
nal, the elements, $1 : j = 1,***,m, of (4)1. must satisfy
j=1
This is equivalent to saying that (0)1. determines a point
on an mdimensional hypersphere of unit radius. If 4 is to
have invariant measure on R(m x m), then the measure on
(0)1. must be
fL(Q) 3 s~l1
m (2.1.2)
= otherwise ,
where
S. (r) = 2aj/2rj1./T(j/2 (.13
 17 
18
is the surface area of a hypersphere of radius r in j
dimensions.
Now fix the elements in the first row~ of 9, and
consider any other row~, say the second rowj for convenience.
Due to the orthoo~nality of 4, the elements, O2 : j = 1,
***,m, of (0)2. mUSt Satisfy
I $1]#2j = 0,
j=1
and (2.1.4)
j=1
This implies that (0)2. determines a point lying on an
(m 1)dimensional hypersphere of unit radius which is
orthogonal to the vector (0)1.. Since 4 is to have invari
ant measure on R(m x m), the measure on (b)2. COnditional
on (4) must be
fl( ~~ ~ )2 T1]=S 1(1 if (2.1.4) is satisfied;
m1
(2.1.5)
= otherwise.
Continuing in thiis manner, fix the elements of
10)1,( 2.,**,()i,.,and consider the ith row,()
of 0. Since 0 is to be orthogonal, the elements, 4 ..
j = 1,***,m, of (0). must satisfy
19
j j ij =
j=1
j=1 i1j1
0,
0,
(2.1.6)
0,
and
j=1
This implies that (0)i determines a point on an (mi+1)di
mensional hypersphere of unit radius which is orthogonal to
the mutually orthogonal vectors (4)1.,(4)2.,***,(4).
11,.
Since @is to have invariant measure on G2(m x m), the measure
on (0). conditional on (0)1 ,(4)2)** ). i 1
.' i1,.
m 1+
if (2.1.6) is satisfied
; (2.1.7)
= otherwise.
Hence the measure on Q is
S1
f(0)= II g (
]=1 j
if QQ' = I;
(2.1.8]
= otherwise.
f (4). 10)z.,(s)2.y
1.
20
Using (2.1.2), we can write the measure on 4 as
j=1 2,j1
= otherwise.
(2.1.9)
In the derivation of the moments of T, we shall
need certain moments and product moments of the elements
of 0. To this end, for any m x m matrix, C, of nonnegative
integers, define
m m c..
J(C) =!n n: H H
ij
i=1 j=1
(2.1.10)
These are the "orthospherical integrals." We shall say
that J(C) is of order s, where
m m
s = C c...
i=1 j=1 II
(2.1.11)
2.2 Some Relationships on the Orthospherical Integrals
Let P and Q be fixed m x m orthogonal matrices and
define an m x m random matrix, 0, by
o = P~QQ,
(2.2.1)
m m c..
= H H 0 1 f(0) de.
I n(mxm) i=1 j=1 i
 21 
wherein 4 is random wdith invariant measure on R(m x m). Then
GeR(m x m) and, since it is obtained from Q by an orthogonal
transformation, O also has invariant measure on. R(m x m).
Now~
m m c..
J(C) =Ir ET H .. 11
i=1 j= 13
m m m m Ic
= Ei H H iaaB (2.2.2)
i=1 j=1 a=1 B=1
By suitable choice of P and Q, (2.2.2) yields a number of
interesting relationships on the orthospherical integrals.
In particular, suppose that P is obtained from the
m x m identity matrix, I, by some permutation of the rows
or columns of I, and that Q is similarly obtained by a
possibly different permutation. P and Q are therefore
orthogonal, and clearly they have one and only one element
in each row and column equal to unity and all other elements
zero. That is,
p.a = 1 if al = i. for some i.
= if a / i., (2.2.3)
where i. differs for each i, and
q.=1 if B = j for some
83 j j
= if 6 f j; (2.2.4)
22
*
where jj diiffers for each
j. Now~
m m
Z Pi/aB 9j
a=lf1 1
(2.2.5)
1j
so that (2.2.2) can be written as
c..j
(2.2.6)
# a
1. ,j
1 j
with 4 the elements of 6, where 6 is defined in
i. ,
Sj
(2.2.1). e may rewrite (2.2.6) as
(2.2.7)
with c. ", the elements of C*:(m x m) defined by
i j
(2.2.8)
C* = P'CQ'.
That is, C* is obtained from C by a permutation of the rowl~s
or columns of C. Since by definition
J(C*]=E 4,
i=1 j=1m m i Ci ]
(2.2.9)
then
(2.2.10)
J(C) = J(C*;).
The permutation matrices, P and Q, were obtained arbitrarily
Thus, for
so that (2.2.10) holds for all permutations of C.
example, with m = 4
m m
J(C) =t E H
i=1 j=1
J(C) =E H H 4 1,
i = 1 j=1mr
 23 
2 01 0 0 00 0
0 0 00 0 3 00
0 3 00 1 00 2
0 0 00 0 00 0
3 00
I O1 OO OO2 0OO
and so on.
Equation (2.2.8) can be used to simplify the
notation by displaying only the smallest rectangular array
containing all of the nonzero elements of C, along with an
index, m, to denote the dimensions of 4 and C. If C is the
smallest rectangular array containing all nonzero elements
of C, then
J(Cc) = J(C). (2.2.11)
For example, with m = 4,
2 00 1
S21 0 0 00 0
0 030 03 0
0 0 0 0
 24 
and so on.
Let P and Q be diagonal matrices whose diagonal
elements are either +1, so that P and Q are orthogonal.
Then from (2.2.2) and (2.2.11),
J (C) = p 3 1
m _r (ii jj) ij
i= =m m c I
=J (C) HI HI (p q ]
mi=1 j=1
m C. m C.
= J C Hp.'Hq (2.2.12)
i=1 3=1.
1i. 1]
j]1I
and (2.2.13)
c c...
*)i=1 13
Suppose that ci. is an odd integer for some iEC1,***,m).
For convenience, assume that cl. is odd. Setting
pii = 1,
pi = +1: i = 2,***,m, (2.2.14)
q. 1: j = 1,***, m,
25
equation (2.2.12) gives
Jm(C) = Jm(C). (2.2.15)
It follows that
Jm(C) = 0 (2.2.16)
whenever any row~ or column of C sums to an odd integer.
Thus, for example,
J =I 0,
and
J2 4 2 0.
By considering other special matrices, P and Q,
many other relationships can be obtained on the orthospheri
cal integrals. For example, choosing
1 1
1 1
(0) I
and (2.2.17)
Q =I,
 26 
equation (2.2.1) yields
and (2.2.18)
622 = (12 ~22 *
Hence
2 0 = E(61128222)
= ES(011 + ~21 2 912 "2
(2.2.19)
Expanding (2.2.19) and making use of (2.2.16) gives
2 J = J (,2) J (2.2.20)
15m 11 LO 2m27
Thus J,1 ( ) can be evaluated once J (2,2) and J (2 0)ar
known .
Other relationships can be obtained by simply
making use of the orthogonality of Q. Some examples are
presented below.
i.~12 = 2 2 j
j=1
=~~ i + 1 $11 $1. (2.2.21)
j=2
27
Taking expectations in (2.2.21) yields
Jm(2) = Jm(4) + (m 1) Jm(2,2).
i=1
= Oil 12~ +11~12 ~i2 .
i=2
(2.2.22)
(2.2.23)
Taking expectations yields
J (2,2) = J (2,4) + (m 1) J (2 2).
(2.2.24)
3. O = C q2
j=1
~11 21 1 2'.l
j=1
m
= 11212 21 ~ 11 21 12.
j=2
(2.2.25)
Thus
Jm(2,2) + (m 1) Jm( : ) = 0.
(2.2.26)
Such relationships greatly simplify the evaluation
of the integrals, since they reduce the number of integrals
which must be evaluated directly. Direct evaluation of the
orthospherical integrals can be tedious, but a relatively
easy method is presented in the nexct section.
 8
2. 3 Evaluatic~z on 3 rthoschearical rinteg7ra~
The orthospherical integrals which need to be
evaluated directly may be evaluated by a number of methods.
A polar transformation on the elements of Q may be used
when the integrals involve only the elements in a single
row~ or column of 9. This method becones extremely cumber
some, however, wLhen elements of more than one rowd or column
are involved. Perhaps the simplest method is one which
depends upon the following representation of 0.
Let X be an m x m matrix of random variables and
suppose
X Ynxm((0),I 0 I).(2.1
Define an m x m matrix R to be a lower triangular matrix
satisfying
XX' = RR'. (2.3.2)
Equation (2.3.2) can be writteni as a system of equations
onth elemennt of' I and R.
2 2
x2 = ral
j=1
x ,Ixi = r"iril: i = 2,***,m,
j=1
2_2 r22
x =r21 i2
j=1
29
j 1 xi = r21ril + r22ri2: i = 3,***,m,
and
m m
j=1 m3 j=1 m)
If the diagonal elements, r i: i = 1,***,m, are required
to be nonnegative, then the equations (2.3.3) uniquely
determine the elements of R. Hence, R is the unique lower
triangular m x m matrix with nonnegative diagonal elements
satisfying (2.3.2). We shall call R the lower triangular
square root of XX'.
Define a matrix 9:[m x m) by
X = RO. (2.3.4)
It is to be noted that the elements of X, and hence of R,
are continuous random variables. Therefore, the diagonal
elements, 'i : i = 1,***,m, will be positive with proba
bility one, and hence R will be nonsingular with probability
one. Then 4 is properly defined by (2.3.4) and
4 = R X. (2.3.5)
Using (2.3.2) and (2.3.5), O must satisfy
94' = (R X)(X'R' ')
 30 
=R 1(XX')R'
=R (RR']R'
=I, (2.3.6)
so that a has zero measure except on a subset of G(m x m).
Let f(X) denote the joint density of thie elements of X
described by1 (2.3.1), and P be any fixed, orthogonal,
(m x m) matrix. Since f(X) > 0 for any real X, then
f(XP) > 0. From (2.3.2), and the fact that P is orthogonal,
w~e have
RR' = XX'
= XPP'X'
=(XP)(XP)'. (2.3.7)
Thus the lower triangular square root, Ri, of XX' is identical
to that of XPP'X'. Then
XC = RO ===> XP = R3P. (2.3.8)
This implies that whatever has nonzero measure, then so
does OP. If g(0) is the measure on then
g(0) > 0 for all OEd(m x m). (2.3.9)
The matrix O will be shown to have invariant
measure on ;l(m x m). The joint density of O and R will be
obtained, and it w~ill be shown that the elements of Ri are
oothi iutua~lly innde deni~t; and independents of thle elements
31 
of d. The joint~ density of R and O Iwill be denoted by
h(R,Q). This joint density can be obtained using the
density of X given by (2.3.1), and obtaining the Jacobian
of the transformation from X to R and O defined by (2.3.4)
and (2.3.6). Denot~ing this Jacobian by J(X R,Q), then
h(R,0) = f(RO) J(X +t R,O), (2.3.10)
where f() is the density of (32..1). Direct evaluation
of the Jacobian is cumbersome and the method used here is
due to Hsu and reported by Deemer and 01kin [8].
We seek the Jacobian of the transformation from X
to R and Q defined by
X = RO,
and (..1
QQ' = I,
with all matrices m x m and R lower triangular. Let gX,
6R, and 60 be n x m matrices of elements, (6X) j, (6R) ,'
and (60) j, denoting small changes in the elemensofX
R, and Q respectively. Suppose that the changes bR in R
and 60 in 4 bring about a change dX in XY so that (2.3.11)
is preserved. That is,
X + dX = (R + 6R)(Q + 60)
and (2.3.12)
(4 + 66)(8 + 60) = I.
 32 
Expanding equations (2.3.12) and dropping terms of second
order in the 6's, we find thiat
X + 6X = RG + R(60) + (8R)
(2.3.13)
anid
Making use of (2.3.11), w\e can write equations (2.3.13)
as
aX = R(60) + (6R) 4
(2.3.14)
and
@(60)' + (60) O' = 0.
Hsu has shown that
J(X +t R,0) = J(6X +t 6R,64),
(2.3.15)
wJhere J(6X; + 6R,64) is the Jacobian of the transformation
from 6X to 6R and 60 defined by equations (2.3.14), in
which R and 4 are now.~ considered to be fixed m x m matrices.
For convenience, denote 6X by A, 6R by B, and 64
by C.
Then
J(X +t R,4) = J(A +t B,C),
(2.3.16)
with
A = RC + Be
and
(2.3.17)
CQ' + QC' = 0.
Define m x m matrices A* and C* by
A*: = AQ''
and (2.3.18)
C* = CQ'.
Using (2.3.17) and (2.3.18),
A* = RC*k + B
and (2.3.19)
C* + C*' = 0,
so that C* is an asymetric matrix. Furthermore,
J(A +t B,C) = J(A + A*) J(A* +t B,C*) J(C* +t C).
(2.3.20)
The Jacobians, J(A +t A*) and J(C* C), are the Jacobians
of the transformations defined by (2.3.18). These are
orthogonal transformations and hence,
J(A +t A*) = J(C* +t C) = 1.
(2.3.21)
The Jacobian J(A* *t B,C*) isthe Jacobian of the transfor
mation defined by (2.3,.19). Recall that B = R, and hence
is lower triangular. Direct evaluation of this Jacobian
can be carried out rather easily. It is in fact
J(A* *t B,C*)
8( x~ 21a 2 ** a ** a *ax*a 3 ** *a2
mlm2 mm Im
... a C...2Ja22 *m ., a2a3.. 2
...9 h...am lmf (2.3.22]
r2m 'm1, m
GC
> 10 i .~ . . .
CO *I *i *
I *
OC *M O I
e I
3 *C I **
CO I h h
r l oi o o .. .0 .
N sH
~ E I
O I
*M
* I r
*M~I O
ON
O i
*H
*
CD E
O 
I R * K*
,0 * * * i U 0 * U U *
34
where L is an m(:21)/2 x m(m + 1)/2 matrixe which is
irrelevant to the value of the determinant of the above
matrix, since the corresponding upper right hand partition
of the above aratrix has all its elements zero.
The Jacobian J(A* B, C") is therefore
m
J(A* B,C*) = Hr..(23.3
i=1
Equations (23.516), (.3.2.0), (2.3.21) and (2.3.23) imply
mi
J(X +t R,0) = Hi r. 2.3.24)
i=1
The joint density; of R and Q is then obtained from (2.3.1),
(2.3.10) and (2.3.24).
m2/2 1 t R m mi
h(R,Q) = (2x) exp y tr RR Hq
on
~EO!a x m), (2.3.24)
r.. > 0: i = 1 *m
and
m < ri < +m: i = 2 ,**,m; j = 1,***,i1.
This density factors into
m m i1
h(R,Q) = h1(3) Hh.(rj= ] h.r
i=1 = =
(2.3.25)
 36 
hl(@) = j 1 on @ER(m x n),
(2.3.26)
mii1 2 lmi 1
hi(r..) = Ir( ]i 2 r.exp  r..
on r.. > 0,
and
h..( I = 2)ep 1 r.." on m < r.. < +m.
1113 i 3
Thus 0 has invariant measure on n(m x m), and its elements
are independent of the elements of R. Furthermore,
r.. ^ (0): i = 1,***,m,
and (2.3.27)
r.. ( ) i = 2,***,m; j = 1,***,i 1.
Using (2.3.4) along with the distributional
properties of X, R and 4, the orthospherical integrals
can be evaluated. Some examples follow.
1. xii = raz~11. (2.3.28)
Hence,
E(XII ] = E(rix 011 )
= Er112 ) J(2). (2.3.29)
(2.3.1) and (2.3.27)
Equations
imply that
E(X112)
and
(2.3.30)
E(r112 m
Substitution into (2.3.29) yields
1
J (2) 
(2.3.31)
2. E(x11 ) = E(rix 11 )
=E(r11 ) E(011 )
= E~r11 ) Jm(4).
(2.3.32)
But
E(X 1 ) = 3
E(r11 ) = m(m + 2),
and
(2.3.33)
implying that
3
Jm(4) =mm+2.
(2.3.34)
(2.3.35)
3. X12 = 11112.
From (2.3.28),
E(XII2X122) = E(r11 Oil #122),
(2.3.36)
and hence
3S
E(x11 ) E(X12 ) = E(rix ) E(011 412 )
= E(r11 ) Jm(2,2).
(2.3.37)
Since xll and x12 are independently distributed as N(0,1),
1
Jm(22 m(m + 2)
(2.3.38)
The value of Jm(2,2) could have been obtained from
the relationship given by (2.2.22). WIe have
Jm(2) = J (4) + (m 1) Jm(2,2).
(2.3.39)
Substitution of (2.3.31) and (2.3.34) into (2.3.39) yields
9 = m(m + 2) + (m 1) Jm(2,2),
J, (2,2) 1 Z.
(2.3.40)
(2.3.41)
so that
Other orthospherical integrals may be evaluated
in a similar manner. The orthospherical integrals up to
order eight, necessary in order to obtain the first four
moments of T, are contained in Table 2.1.
(The integrals up to order eight not included in this
table are equal to zero)
39
TABLE 2.1
Orthospherical Integrals
A1 = m (m +
a, = m(m +
A 3 = m (m +
2) (m + 4)
2) (m + 4) (m + 6)
Order 2:
1
J (2)
Order 4:
J (4) 3~i
1
J (2,2) 1
_(m+1)
(m 1) 6
(m 1) A
J(2 01
J)( :
Order 6:
15
J (4 ,z~ 2)
40
TABLE. 2. 1Cont~inued
3 (m+ 3)
~(m 1~) 2
3
(m1) 6
1
62
Jm( a _
Jm( o 2
(at~~z
J(2 2' mi
mo 2 (m1]62
(m2+3m2)
S(m2) (m1)n2
S(m1) 62
= (m+ 2)
(m2) (m1) A2
2
S(m 2) (m 1) a2
Order 8:
105
As
Jm s
J (6, 2)
Jml )
15
A 3
_ 5(m+5)n
I2 0 0 j
002
J I 2O 11 1
011
/110 j
011
41
TABLE 2.1Confir~ued
Jm(4,4)=
Jm4 o) 9(m+3)(m+5)
me 4 (m1)(m+1)A3
J (4 22) ~mi
(42 3(m+ 1)
2,( _) 3(m+3)
mo 2 (m 1) a3
S3(m+5)
= 3(m+3)(m+5)
S(m1)(m+1)a3
3(m3+8m +13m2)
= m2)(m1)(m+1)n3
(m1)n,
 3(m +7m+14)
[m2)(m1)(m+1)n3
(m1)A3
J, O 2O 2
J (o 4O2
I4 0 0 I
J/0 0 2
moI 31 3
 9(m+3)
(m1)(m+1)A3
Jm(o (m1)A
I~
42
TABLE 2.1Continued
_3(m+ )
m, 31 1
= 3(m2+5m+2)
S(m2) (m1) (m+1)A3
J (2,2,2, 2) ~~
2 2 2 mln
m 0 0 3
J(2 2) (m +4m+15)
m2 2 (m1)(m+1)n3
(m+3)2
S(m 1)(m+1)n3
J 2O 2
J 2O 2
0 0 (m+j) (m+5)
2 2 (m1) (m+1)a3
2 2 0 (3 62+m6
J
ma 2 0 (m2) (m1) (m+1l)n3
22 0) (m+3) (m+5)
Jmioo 2 =(m1) (m+1)n3
to a 2
2 2 0 0 (3+8m2+13m2)
J [00
o 2 co0 (m+3)(m3+4m 11m2)
0 0 0 2 m3~~7r~'7i~
3 Io
J 0
m 1 10
 43 
TABLE 2.1Continued
1 m
( m1)(m+1)n3
J, 2 1
Jm 2O 2
_ 1
(m1)63
(m2+7m+14)
S (m2)(m1)(~m+1]63,
 (m2+3m+6)
S(m2) (m1) (m+1)n3
 (m+3)
S(m1)(m+1)63
(2+5m+2)
(m+3)
S(m1)(m+1)a3
(m3+6m2+3m6)
(m )(m 2)(m 1) m+1) 63
(mnJt1) m1)6
Jmo 2O O
1 1
1 1
Jm( 11
(m +6m+16)
2 2 00
J = 01
mI0 I
1 201
2 0 11
J.[ 0
mooI 20
0 0 11
1J~ 1 111
1~ 1 0 O
80011
CHAPTER III
M\OMENTS Of HOTELLING'S GENERALIZED T_
3.1 Introduction
In Chapter I we defined Hotelling's generalized T 2
(see equation (1.1.11)) as
v1
T 2 = 92 r v.S,'v
O  1i
i=1
= 2 tr S S21,
where, under normal assumptions on the underlying random
variables, SI and S2 are n x independent random Wishart
matrices with
and (3.1.2)
S2 ami, 2/0) .
It is assumed_ that T7 is positive definite. In view~ of the
invariance of the general linear hypothesis and T 1 under
linear tranisformations (see Section 1.1), w~e may simplify
the derivation of the moments of T "
Since i' is positive definite and
i=1
 44
 45 
(see equation (2.1.13)) is nonnegative definite, there
exists an m x m matrix, G, satisfying
GVG' = I
and(.1)
GMM~'G' = A
where A is an m x m diagonal matrix with diagonal elements
X1,'**,Xm defined in (1.1.23). W~e may therefore assume,
without loss of generality, that in our definition of T *
given by (3.1.1), the column vectors q;: i = 1,***,vl and
the m x m matrix S2 are independently distributed with
7. N (5 I): i = 1,***,v l, (3.1.5)
and
S2 W m(I,v2,(0)j, (3.1.6)
where ii: i = 1,***,,v stisfy
11
i.i = A.(.1 7
i=1
W',rite
Y = ( x,** ,v ](3.1.8)
and
so that Y is an m x vl ma*trix of random variables and Z is
an m x us matrix of means, and
46
Y Nmx zE I0 I .( .0
From (3.1.1) we may write
T = V2 r v 'S2lv
i=1
=V2 tr Y'S2 Y. (3.1.11)
In this chapter, a method is developed for evaluating
the moments of T 2, or,more precisely, the moments of
T 2
under the assumptions (3.1.6) and (3.1.10) with T_ defined
by (3.1.11). The first tw~o moments, E[T] and E[(T) ] are
found explicitly. It is noted that the rth moment of T
exists, provided v2 > 2r +t m 1.
3.2 A Method for Evaluating the Moments os T_
We have
T
V2
=tr Y'S21Y,(. .1
with Y and S2 distributed as in (3.1.10) and (3.1.6). Let
dl,***,dm be the solutions of the determinantal equation
1S2 GII = 0 (3.2.2)
so that di,***,dm are the latent roots of S2. Define D to
be an n x m diagonal matrix ,ith
 47 
D = dl 0 * 0 (3.2.3)
0 d2 * 0
0 0 * d
If R. is an mdimensional random, normed eigenv;ector associ
ated with d., then
LS2L' = D, (3.2.4)
where L is the m x m random matrix satisfying
L = (1,***,E ) (3.2.5)
and
LL' = I, (3.2.6)
so that L is orthogonal. L, and D3 are uniquely defined by
(3.2.4), (3.2.5) and (3.2.6) if we require dl > d2
> dm and the elements of 1j: j = 1,***,mof the first
row of L to be positive. The joint density of L and D
will be obtained, and it will be shown that the elements
of L are independent of the elements of D. The Jacobian,
J(S2 + L,D), of the transformation from S2 tO L and D will
be obtained by the method used in Section 2.3.
Let 6S2, bL, and dn be m x m matrices of elements
(6S2) ,, (SL)i and (gD)i.. representing small changes in
 48
the elements of S2, L, and D respectively. We require that
(L + 6L] remain orthogonal and(D + 6D) remain diagonal.
Suppose that the changes 6L, in L, and dD, in D, cause a
change 6S2 in S2 SO that (3.2.4) and (3.2.6) are preserved.
Then
S2 + 6S2 = (L + 6L)'(D + 6D)(L + 6L)
I = (L + 6L)(L + 8L)' (3.2.7)
By expanding (3.2.7) and dropping terms of order greater
than one in the 6's and using (3.2.4) and (3.2.6), wie have
6S2 = L'D(6L) + L'(6D)L + (6L] DL,
and (3.2.8)
0 = L(6L)' + (6L)L'.
As in Section 2.3
J(S, t L,D) = J(6S2 + 8L, 6D) (3.2.9)
where J(6S2 + 6L, STD) is the Jacobian o'f the transformation
from 6S2 to 6L and 6D defined by (3.2.8), in which L and D)
are now assumed to be fixed m~x m matrices. For convenience,
denote bS2 by A, GL by B, and 8D by C. Then from (3.2.8)
and (3.2.9] we seek the Jacobian J(A4 + B,C) defined by
A = L'DB + L'CL + B'DL
and (3.2.10)
0 = LB' + BL'.
 49 
Define m x m matrices A* and B* by
A* = LAL'
and (..1
B* = B,'.
Using (3.2.11), equations (3.2.10) may be written as
A* = DB* + C + B*'D
and (3.2.12)
0 = B*' + B*,
so that B* is an asymmetric matrix. Then
J(A +t B,C) = J(A +t A*) J(A* +t B*,C) J(B* *t B) (3.2.13)
where J(A4 + A*) and J(B* B) are the Jacobians of the
transformations defined by equations (3.2.11), and
J(A* +t B*,C) is the Jacobian of the transformation defined
by equations (3.2.12). Equations (3.2.11) represent
orthogonal transformations and hence
J(A +t A*) = 1
and (3.2.14)
J(B* *t B) = 1.
Equation (3.2.13) then reduces to
J(A +t B,C) = J(A* + B*',C).
(3.2.15)
 50 
Direct evaluation of J(A*' i B*,C) is straightforward.
Recall that C = SD and hence is diagonal, B" is asymmetric
and A* is svaretric.
a(all~a22****a *a21*ast~as****a *~a
J(A* *t B* C) =mm ml m2
3(c 1 c 22  c b 2 1*b a "b 32****b *b *
mm ml m2
...a *)
...b *) (3.2.16)
m,m1
Laying out the matrix of derivatives in tubular form, we
have
I
51
So o ***C o
I
CO O
Ocl
0 0
***D
UCA i:i
***D f
U U
 52 
The Jacobian J(A*" B*',C) is the absolute value of the
determinant of the above matrix and is
Hi (d. d.].
i
J(A* B*,C) = H
(3.2.17)
Thus, from equations (3.2.9), (3.2.15) and (3.2.16),
Hm (d. d.].
J(S2 + L,D) = H
1
(3.2.18)
From (3.1.6), the density on S2 is
j=1
x S2r2m8/2
(3.2.19)
on S2 positive definite.
The joint density of L and D,
g(L,D), is, from equation (3.2.4)
g(L,D) = f(L'DL) J(S2 + L,D)
(3.2.20)
where f(*) is the density of (3.2.19). Recalling that L
is orthogonal, we find that
g(L,D) = 2 92/2 .m(m1)/4 m pv
3=1
x H d (2m1/2
j=1 3
ex ( d) H (d.d .]
3
(3.2.21)
ep t S2
53
on L satisfying LL' = I, R,j > 0: j = 1,***,m and dl > d2
> *** > dm > 0. The density of (3.2.21) factors into
g(L,D) = gl(L) g2(D)
(3.2.22)
where
j=1
if LL' = I, RA > 0: j = 1,***,m
= otherwise,
(3.2.23)
and
.'"/2
exp( d. H d .
2 1l
if dl > d2 > *** > dm > 0
= otherwise.
(3.2.24)
Hence the elements of L are independent of the elements
of D.
Suppose that K' is any m x m random orthogonal
matrix satisfying
KS2K(' = D.
(3.2.25)
Write
( ki,k 2, ***, k ]
(3.2.26)
821D) = m22 H r( 2 t
j=1
x H d.(2m 10/2
j=1 3
54
whee k is an mdimensional, random, normed eigenvector
associated with d. Comparing (3.2.26) with (3.2.25), we
have
1
P~lk. = i 1. 2 :
(3.2.27)
and
P(ki i] = i=
The measure on K; is obtained from
1,***,m.
(3.2.23) and (3.2.27).
It is in fact
m.2
2 =
if KK' = I
(3.2.28)
= otherwise
so that K has invariant measure on n(m x m) and is indepen
dent of D.
With Y defined by (3.1.10) and Kt defined by (3.2.28),
let Z be an n x vi matrix of random variables satisfying
(3.2.29)
Z = KY.
From (3.1.10) the distribution of Z, conditional on K;, is
ZIK ^ Nmy (KE,I O I).
(3.2.30)
The unconditional density on Z can be obtained from (3.2.30)
and is
 55 
f(Z jKER(mm) mvl/3 7c 1
f(Z) =(2x) ~ x 2 tr(ZK >)(ZKE)'
x dK. 3.2.31)
j=1 2xr
It should be noted that since Y and S2 are independent, Y,
Ki and D are mutually independent. This implies the inde
pendence of Z and D.
Using equations (3.2.1), (3.2.25) and (3.2.29), we
have
T = tr Y'S21Y
=tr(Z'K)(K'D'K)(K'Z)
=tr Z D'Z
m vl z..
= (3.2.32)
i1= 3=1 1
Wrri ting
V1
u. 2 =Z..2,
S i=1 1
m u.2
T = C.
i=1 1i
(3.2.33)
then
(3.2.34)
This representation of T, along with the results of
Chapter II, and the distributional properties of Z and D,
can be used to evaluate the moments of T.
 56
3. 5 The FSirst Two Mlomen~ts of T
Using the representation of T given in equation
(3.2.39) the first twio moments of T are
E (T) = C E d:
1= 1
and
m U.2UiU.
i=1 di fj"1
= ~ E~ t +  liu~) F
i= d Ir did
(3.3.2)
From equations (3.2.30) and (3.2.33), u.2 COnditional on
K is the sum of vl independent normal random variables.
Thus
u.2 K I^ X (KA').. : i = 1 m (3.3.3)
where from (3.1.7)
57
A = EE'.(3.3.4)
For convenience, let
Y. = (KAK')..
= k.. A.i (3.3.5)
j=1 1
since A is diagonal. Then
u 2K ^ X 12 Y ]: i = 1,2,***,m. (3.3.6)
Furthermore, conditional on K, u.2 and u.2 are independent.
Hence, using (3.3.6)
E(u =) E E (ui2
K u2 K
= Elv1 + Y ), (3.3.7)
E(U = E E (ui4
K ui2 K
=EK va(vi + 2) + 2y.(vl + 2) + y'n (3.3.8)
and
E(u 2u. = E E Iu u.21
SK iu.,uIK 13
=E E (u.2 E iuj
K u. K u. K
=EI vz' +( vj Y. 7 Y. (3.3.9)
58
Recalling that K; has invrariant measure on R(m x m), w~e may
use the results of Chapter II to evaluate the moments of
yi and the product moment, EKy(viY ] where Yi is defined in
(3.3.5).
E (y i) =E k..2X
K' K j=1 11 3
=J (2) X.
= A, (3.3.10)
mj=1
E(y2 => E k. 2Xj
K K j=1 3
= Ei 4X k." +C~i~i k. k .2
Kj=1 133 1 3
=J (4) 1 .2+J (2.2) A~ R .
m j=1 3 mE
j=1 m1+~ f
and
m m
= ~ i E .2 k 2 2;
K . =1 1 l 1 t i Et
59
1 (m+1)
mTmTI A 1a T m1)~nm~m+2 RR
Substitution of equations (3.3.10), (3.3.11) and (3.3.12)
into (3.3.7), (3.3.8) and (3.3.9) yields
i=1
(3.3.13)
E~u] =vl(vl + 2) + (2vl + 4) 1 X
j=1
+3 2 1 LA xxj
m(m+2) j m(m+2) j
j=1 i j
(3.3.14)
and
m
( 2v X.'
1 3 j=1
+ ~m+1~inZ Z .. Z ij
(m1m~m2) 1
(3.3.15)
Upon substitution of (3.3.13), (3.3.14) and (3.3.15) into
equations (3.3.1) and (3.3.2), the first two moments of T
are
]
(3.3.16)
and
m m
F(T) = i + 1 E C
=1 1=1
60
i=1
+ m
+~+2 mC .j2 + 1 I
m~+)j=1 3 im2)1
m m
x c" 1 2 ] I 2vi X
i=1 di j=1
1 m+1
m(m+2) 1 =i 1
x d1 (3.3.17)
Evaluation of the terms
m m
=1 1i1 1
and
involve some special problems. TThese w~ill be dealt w~ith
separately in the next section. W~e list these expectations
here, however, in order to obtain the mnoments of T:
E =v m1 (3.3.18)
i=1 1
61
m(v21)
(V2m)(V2m1)(v223)
(3 >.19)
and
S1 m(m1)
c:7 [v ) 2 .
ir)1
(3.3.20)
Substituting these expectations into equations (3.3.16) and
(3.3.17), we obtain
myi + A.X
B(T) = 1(3.3.21)
v2m1
and
=mv12(my2 m22m+2) + 2my (v21)
+ (2vl(my2n22_~2) 2 ~1~j
i=1
m
i=1
(~.3.23)
Numerous checks are available on the moments of T.
The distribution of T, and hence its moments,
in the following special cases. If m! = 1,
is known
T ^ '
x yo)
L2
(3.3.23)
m
~ d 2
1 1
i=1
 2
Putting m = 1 in equations (3.3.21) and (3.3.22), the first
two moments of T reduce to the first two moments of a ran
dom variable distributed as in (3.3.23). In the case of
v1 = 1, T2 reduces to Hotelling's T' defined in (1.2.1).
Hence when vl = 1,
T = 7.
(3.3.24)
The distribution of Hotelling's T2 iS knOwn to be [14]
V2X 2IX
T2 h
X 2 *
(3.3.25)
It should be noted that when vi = 1, there is at most one
nonzero solution of
A1 XII = 0.
(3.3.26)
Then w~hen vl
x 2_ (0)
V2 e
(3.3.27)
Putting v? = 1 in equations (3.3.21) and (3.3.22), we see
that the first twio moments of T reduce to those of a random
variable distributed as in (3.3.27). The limiting distri
bution of T 2, as V2 tends to infinity, is known to be
(see [15] or [26])
lim T 2
o
\12+m L
(j3..28)
f
1 E
X 2 1. .
my 1
1 1=1 o
63
In equation (3.3.21), if we multiply byI v2 and take the
limit we obtain
lim
i m)O T =' mvl + mC ..i (3.3.29)
Similarly, multiplying (3.3.22) by v22 and taking the
limit, we have
E li T =m2 2+ 2mvi + 2(muz+2) m. + 1
v2+im o =1 i=
(3.3.30)
Equations (3.3.29) and (3.3.30) are the first twso moments
of a random variable distributed as in (3.3.28). As a
final check on the moments of T, it is known that the
distribution of T is invariant under the transformation
(see [1])
(m,vl,V2) t (Vl,m,v2+v2m).. ..1
Simple substitution shows that the moments of T given in
(3.3.21) and (3.3.22) are in fact invariant under this
transformation.
In the null case when X1 = X2 = *** = Am = 0, the
moments of T reduce to
E(TliX = ** = 0] my' (3.3.32)
64
and
(V2m) (v2m1) (v2mn3) E((T) (13 = *** = Am =
=mv1 (mv2m22m+2) + 2mvl~v2) 3.3.33)
3.4EauaonoCeanExeatosUd
in Obtaining the Moments of T
In Section 3.3, we listed the values of E 
E d7and E a .
~1= 11 1
in order to evaluate the moments of T (see equations
(3..1),(3.3.19) and (3.3.20)). Wfe derive these expec
tations here. Recall that di,***,dm are the latent roots
of S2. Thus
= tr S21', (3.4.1)
dl = tr IS21 2, (3.4.2)
i=1 i
and
1 1 = (tr S21 2 tr(S2~']'. (3.4.3)
We shall need some results in matrix algebra and
some w\ell known properties of the Wdishlar~t random matrix
(see [1] or [32]). Let A be an m x m nonsingular matrix
and partition A? into
65
A21
'
~22
(3.4.41)
where A21 is a p x p
matrix with p1 + p2 = m.
Then
(3.4.5)
Suppose that B is an m x m matrix satisfying
B = A'.
Partition B in the same manner as A,
(3.4.6)
(3.4.7)
Then
B1II AII A12 ~221 21)
and
(3.4.8)
B22 122 A21All1Ax2)
With A defined and partitioned as above, suppose
that
A Wm(Iv,(0)].
(3.4.9)
Then A1 and A22 are independently distributed and
JA = JAlljA22 A21All A12 ,
= A22 )11 12 22 121*
B11 i B12
B21 : B22*
 66 
q p Iv,0] i = 1,2. (3.4.10)
Also, A.. is distributed independently of [A. A.A ,..'A..
w~ith
33 31. i 11 134 :J
Furthermore, (A.. A..iA..1A..) is independent of
4..A.. 'A.. ^ Wz (I,p ,(0)). (3.4.12)
It is important to note that the W~ishart distribution w~ith
m = 1 on v degrees of freedom is identical to the X2dis
tribution on v degrees of freedom. Thus if
then (3.4.13)
u ^ X 2(0).
With these results, we can proceed to evaluate
E(tr S2 ', E(tr(S2 ')2} and {(Itr S2 ]2 tr(S21)2}.
For convenience, we shall work with the corresponding
expectations in terms of A, where A is defined by (3.4.9).
We require E(tr AL) E tr(A L)2 and E{(tr A ~) tr(A1 2
Let
B = A'. (3.4.14)
Then, by symmetry,
67
Eltr A '
=E(tr B)
m
i=1
b..~
(3.4.15)
=mE (b 11).
From equation (3.4.8) with pi = 1 and p2 = m 1,
bii = (aii A12A221A21)
(3.4.16)
Thus using (3.4.11) and (3.4.13)
1
b Xv m+1OT
13.4.17)
and hence
E(tr A1) = m .
(vm1)'
The evaluation of E{(tr X')2 tr(A1)"}
(3.4.18)
is
similar. By symmetry
E((tr A')2 tr(A ') ]
= ((tr B)
tr~(B>)
bij..
E (b Ilbez
m m
C 
=13=1
(3.4.19)
=E ;i m
i1
=m(m1)
b..1 )
 b 2 )
 68 
With pi = 2 and p2 = m 2 and using (3.4.8), wse have
bllb22 b122 = /B1zi
= [A11 Al2A22 121 1.
(3.4.20)
Therefore,
(3.4.21)
El(tr A1 2
Denote (A 1 Al2A 22 A21) by
(All Al2A22IA21)
=C11
C21
C12
C22 ,
(3.4.22)
so that from (3.4.11)
C W 2II,vm+2,(0)].
(3.4.23)
Now from equations (3.4.5), (3.4.10) and (3.4.11),
(3.4.24)
w~ith
tr(A1 2) = m7(m1) E I.lA1A2
2
IC = cii c22 C12
C11
 69 
cil ^ Xym+22(0)
and
(3.4.25)
This implies that
E((tr A')2 ir1 2 = m(m1) E 1~ Pi,,12
m(m1)
= (v m)(v m 1)
(3.4.26)
The evaluation of E~tr(A4 )2) can be carried out
using (3.4.26) and evaluating E(tr A )2.
metry we have
E(tr A')2 = E(tr B)2
i=1
11 i 1 3
1=1Ir
Again by sym
=mE(b112) + m(m1) E(bizb22).
(3.4.27)
From (3.4.17),
E(bli2) = 1 .
(3.4.28)
To evaluate E(bllb22) COnSider (3.4.7) and (3.4.8) with
pi = 2 and p2 = m 2. We have
C2
C22 1 ^ X2( )
c1lv+
70
bll bl2
B11 =
b21 b22
= All Az2A221A~!)1
=C' (3.4.29)
where C is defined in (3.4.22) and distributed as in
(3.4.23). Another application of (3.4.8) yields
bii = cir i
C22
and (3.4.30)
b22 =22 C122 1
11
Thus
c11
blib22 = 11022 i 1 3..1
C11c22
The remarks preceding (3.4.10) along with (3.4.23) imply
the independence of cll and C22. Also, the remarks pre
C22
ceding (3.4.11) imply the independence of cll and C22 1
C11
Hence cll is independent of 11 c'2 )~. Similarly, c22
c11022
is independent of (1 C12 ) and,hence, so is the product
c11C22
cllc22. This implies that
71
22
C11 C22 C11022i
= ~ ~ ~ 1 E 2 (3.4.32)
i 11 i 22)"C C122
C111 C2
C22
The expectations E[ and EI can be evaluated using
(3.4.10), (3.4.13) and (3.4.23), yielding
[C111 vm1
and (3.4.33)
C221 vm1
To evaluate
C11
[ C22
we need to make use of (3.4.12). Let
c122
u2 = c11
C22
and (3.4.34)
C22
c22
From (3.4.12) and (3.4.23), u, and u2 are independently
distributed as
72
u' ^ Xvm,+1 (0)
(3.4.35)
and
u2 ^ X1 (0).
Therefore
C11 1 + U2
C11 C122 U1
22
1
Se(1l,vm+1) *
(3.4 36)
Equation (3.4.36) yields
(vm)(vm2)
(3.4.37)
Substitution of (3.4.33) and (3.4.37) into (3.4.32) gives
E (b 11b22) = (v m) (m [Tvm3), )
(3.4.38)
and wde obtain from (3.4.27), (3.4.28) and (3.4.38)
mm22m+2) .
V m)~v m 1)(v m 3)
E(tr A1)2
(3.4.39)
Finally substituting (3.4.39) into (3.4.26) and solving for
E~tr(A ')2):we find that
(vm)(vm1) (vm3).
(3.4.40)
/3
Putting A = S2 and replacing v by v2 in equations (3.4.18),
(3.4.26) and (3.4.40) we obtain the equations
E(tr S21) = Tm
(3.4.41)
(3.4.42)
m(v21)
(v2m) (V2m1)(v2m3)
E~tr(S21) ] =
and
E((tr S2')Z tr(S2')) = m(m1) (3.4.43)
which are the same as equations (3.3.18), (3.3.19) and
(3.3.20).
3.5 Summary
The representation of T given in (3.2.34),
m u.2
i=1 1
can be used to evaluate higher order moments of T. The
expectations involving the elements u1 ,***,um can be
evaluated using (3.3.6) and the orthospherical integrals.
The orthospherical integrals necessary for the first four
moments of T are tabulated at the end of Chapter II. The
expectations involving elements of D can be found using
simple properties of the Wzishart distribution similar to
those in Section 3.4.
.. 7 4
It should be noted that the second nament of T
depends upon
i= 1
This refutes the claim by MIikhail [27] and Srivastava
[33] that the power of T depends upon Al *, only
through
i=1
CHAPTER IV
APPRIOXIMATIONS TO THE DISTRIBUTIION OF
HOTELLING'S GENERALIZED T z
4.1 Introduction
It was pointed out in Chapter III that the
distribution of Hotelling's generalized Tq is well known
in certain special instances. These are the cases when
m = 1 for all vl and V2; when vi = 1 for all m and V2 > m;
and when v2 goes to infinity for all m and vl. In addition
to these special cases, Hotelling [13] derived the null
distribution of T_ for m = 2, all vi and v2 > 2. With
the exceptions of these fewY instances, the distribution
of Hotelling's generalized To is customarily in series
form in terms of the zonal polynomials as described by
James [21] and Constantine [3]. Constantine [4] gives a
series representation of the distribution of T. His series
is convergent, however, only when [IT < 1. In 1968, Davis
[6] showed that the null distribution of T satisfies a
linear homogeneous differential equation of order m, and
that Constantine's series reduces to the solution of this
equation in the case ITI < i. Davis [7] uses this result
to obtain the upper 5% and 1% points of To'/v1 for m = 3,4
75
 76 
and various values of vl and V2 which he claims are
generally accurate to five significant figures.
In 1954, Pillai [29] suggested that the distribution
of T be approximated by
2 1
m (v2m1)+2 Fmvz,m(v2m1)+2. 411
It should be pointed out that this approximation becomes
exact for m = 1 as well as for the limiting case when v2
tends to infinity~. However, the approximation is not
exact in the case vl = 1, nor is it invariant under the
transformation
10I,V1,V2) 12 ( 1 m~+2 m
In this chapter three approximations to the
distribution of T are considered. Upper 5% and 1% points
are found for each of the approximations in the cases
m = 2,3, and 4 and some accuracy comparisons are made.
4.2 Approximations to the Distribution and
Comparison of Percentage Points of T
In view of the known distribution of T_ for the
special cases mentioned in Section 4.1, a number of approxi
mations to its distribution are proposed for the general
situation. Of course we would like the approximation to
be a simple one in terms of tabulated functions and to
77
satisfy as many known results as possible. Naturally, we
would want any approximation to be sufficiently accurate
for practical purposes. For these reasons, the following
three approximations to the distribution of T are considered:
Approximation 1
T I aX l(Y) (4.2.1)
for some al, v, and y;
Approximation 2
T X aF (Y) (4.2.2.)
*mvi,v2l
for some al and y; and
Approximation 3
T aF (Y) (4.2.3)
.v,v2m+1
for some al, v, and y.
Approximation 1 was chosen in view of the fact
that the limiting distribution of TO2 iS
X 2 1 *i
mv, . 1
It was suspected that this approximation would be particu
larly suited for large v2. AppTOXimation 2 waS chosen
since in the cases m = 1 or vi = 1, the distribution of T
78
is given by (3.3.23) and (3.3.27), respectively. It was
thought that approximation 2 would be fairly accurate
overall. The third approximation was chosen for reasons
similar to, and in hopes of improving, approximation 2.
In approximations 1 and 3, the first tw~o null
moments of the approximating distribution w~ere equated to
the first tw~o null moments of T given in (3.3.32) and
(3.3.33) in order to determine the unknown multipliers,
a, and unknown degrees of freedom, v. In approximation 2,
the first moment of the approximating distribution was
equated to the first null moment of T given in (3.3.32) in
order to determine the unknown multiplier, a. The first
nonnull moment of each of the approximating distributions
was then equated with the first nonnull moment of T given
in (3.3.21) to obtain the unknown noncentrality parameters,
y. Derivration of these unknown multipliers, degrees of
freedom, and noncentrality parameters follow.
Let u be a random variable such that
u aX l(Y). (4.2.4)
The first two null moments of u are known to be
E(uly=0) = av (4.2.5)
and
E(u ly=0) = u'v~v+2).
(4.2.6)
79
Equating these two expressions w~ith (3.3.32) and (3.3.33)
wLe obtain
(v2 1 vl~2m
(v2m (2m (2m3
Smv,(v2 m (2m
() V21)(V +v2m1
(4.2.8)
and
The first nonnull moment of u is given by
E(ulv) = a(v + y).
(4.2.9)
Equating this with (3.3.21) and using (4.2.7) and (4.2.8)
yields
S(v21)(Cvl+V2m1) i
i=1
m
m i=1
(4.2.10)
where the value of v is given by (4.2.8).
Thus approxi
mation 1 becomes
T d aX 2(y), (..1
wh~iere a, v), and y are given by (4.2.7), (4.2.8) and
(4.2.10) respectively.
If u is a random variable such that
u amvl,v2m+1y),
(4.2.12)
 0
then the first null moment of u is
E~u =0)= a(v2m+1).
E~ul~O) = [v2m1)'
(4.2.13)
The evaluation of a by equating (4.2.13) with (3.3.32)
yields
C1v2m+1'
(4.2.14)
Using (3.3.21), (4.2.14) and the first nonnull moment of u,
E(u y) = 0 291 7 2 l
C1mv(v2m1)
(4.2.15)
w\e obtain
m
i = 1
i=1
(4.2.16)
Approximation
2 then becomes
(4.2.17)
m X
T mT Fmvi,v,m1 i .
If u is a random variable with
u av,v,m+1(),
(4.2.18)
the first two null moments and the first nionnull moment of
u are respectively
E(uly=0) = a v21m+), (4.2.19)
[v2m1)
 1 
E(u2ly=0) = 2 (v+2)(v2 m12
( vlyv2lv3'
E(u 7) = a (,~l
(4.1.20)
(4.2.21)
and
A procedure similar to that used for approximation 1 yields
the following values for a, v, and y:
a= mvi ,
(4.2.22)
(4.2.23)
vt = m
Svl+v2 1l
and
m
Y V2m C Xi
vl~v2mvil i=1
i=1
(4.2.24)
It should be pointed out that the value of v given in
(4.2.23) may be negative when m and vl are large, relative
to v2. We W~ill agTSO tO use approximation 3, therefore,
only when the value of v is positive. That is, when
V2 > ml 1+1. Thus, approximation 3 becomes
* mvi v
.2 v2m+lT v ,v2m+1 mV 1' i~CX1
i=
(4.2 25)
 2 
provided v > 0, where
v = v~2O.(4.2.26)
Sv:+v2mvl1
Approximations 2 and 3 become exact in the cases
m = 1 for all vl and V2; V1 = 1 for all m and V2; and in
the limit as V2 goes to infinity. They are also invariant
under the transformation
[m,vl,V2) (Vl,m,Vlfv2m).
Approximation 1 is also invariant under this transformation
and becomes exact in the limit. It is obviously not exact,
however, when m = 1 or vi = 1.
The following tables give the upper 5% and 1%
points of T obtained from each of the three approximations
considered in this chapter, as well as some accuracy com
parisons. In the case m = 2, our approximate percentage
points are compared with the exact percentage points ob
tained from the distribution derived by Hotelling [13] as
well as those found using Pillai's approximation [29].
For m = 3 and 4, comparisons are made with Pillai's
approximate percentage points and those given, by Davis (7].
Hotelling's exact percentage points are reprinted from
[30]. Percentage points for approximation 1 were obtained
using Percentage ~oints; of thie X'2distribution [28]. The
percentage points for approximations 2 and 3, as well as
vt13 23 33 43 53 63 83 125
E 1.4508 .6807 .4429 .3280 .2603 .2158 .1607 .1064
P 1.3966 .6683 .4385 .3261 .2596 .2156 .1610 .1069
3 1 1.4994 .6861 .4446 .3286 .2606 .2159 .1607 .1064
2 1.4979 .6952 .4524 .3341 .2648 .2192 .1631 .1078
3 1.4473 .6820 .4447 .3297 .2619 .2172 .1619 .1073
E 2.1771 1.0071 .6517 .4812 .3813 .3157 .2348 .1552
P 2.0893 .9849 .6424 .4763 .3784 .3138 .2340 .1550
5 1 2.2284 1.0097 .6515 .4807 .3808 .3153 .2345 .1551
2 2.2945 1.0439 .6703 .4924 .3889 .3212 .2381 .1569
3 2.1741 1.0073 .6525 .4821 .3821 .3164 .2354 .1557
E 2.8802 1.3197 .8508 .6269 .4960 .4103 .3049 .2013
P 2.7658 1.2907 .8384 .6202 .4920 .4076 .3035 .2008
7 1 2.9324 1.3193 .8488 .6253 .4949 .4094 .3043 .2010
2 3.0819 1.3829 .8817 .6454 .5084 4192 .3101 2038
3 2.8828 1.3212 .8521 .6279 .4970 .4111 .3055 .2017
E 4.9409 2.2260 1.4243 1.0451 .8247 .6809 .5046 .3323
P 4.7566 2.1806 1.4050 1.0346 .8182 .6765 .5023 .3314
13 1 4.9920 2.2157 1.4169 1.0402 .8213 .6784 .5032 .3317
2 5.4135 2.3783 1.4998 1.0903 .8549 .7024 .5173 .3383
3 .. 2.2342 1.4288 1.0480 .8268 .6825 5056 .3329
 3 
TABLE 4.1
Upper 5% points of T
(m = 2)
V1 13 23 33 43 53 63 83 123
E 2.2704 .9850 .623s9 .4558 .3589 .2959 .2189 .1440
P 2.0541 .9350 .6029 .4445 .3519 .2912 .2165 .1430
3 1 2.1429 .9740 .6068 .4460 .3525 .2914 .2164 .1428
2 2.4103 1.0250 .6438 .4675 .3666 .2820 .2222 .1456
3 2.3025 .9953 .6293 .4593 .3613 .2977 .2201 .1447
E 3.2642 1.3893 .8739 .6361 .4998 .4114 .3039 .1995
P .9624 1.3218 .8456 .6208 .4902 .4050 .3004 .1980
5 1 3.0447 1.3287 .8467 .6209 .4900 .4947 .3001 .1978
2 3.5800 1.4807 .9167 .6609 .5159 .4228 .3104 .2025
3 3.3335 1.4100 .8840 .6422 5038 .4143 .3056 2003
E 4.2220 1.7736 1.1099 .8058 .6320 .5197 .3833 .2513
P 3.445 1.6922 1.0764 .7878 .6208 .5121 .3792 .2495
7 1 3.9124 1.6911 1.0735 .7855 .6191 .5108 .3783 .2491
2 4.7468 1.9223 1.1796 .8461 .6583 .5382 .3939 .2561
3 4.3327 1.8065 1.1258 .8152 .6382 .5241 .3859 .2525
E 7.0198 2.3804 1.7842 1.2879 1.0065 .8255 .6069 .3964
P 6.4298 2.7618 1.7373 1.2634 .9915 .8155 .6015 .3942
13 1 6.4443 2.7339 1.7208 1.2530 .9844 .8104 .5985 .3928
2 8.1696 3.2149 1.9432 1.3809 1.0675 .8687 .6317 .4078
3 .. 2.9509 1. 8182 1.3079 1.0196 .8349 .6123 .3989
 4
TABLE 4.2
Upper 1% points of T
(m = 2)
V2
v1 30 40 50 60 80 100 200
D 1.0385 .7348 .5682 .4631 .3380 .2661 .1289
P 1.0097 .7201 .5595 .4574 .3351 .2644 .1286
5 1 1.0361 .7329 .5668 .4621 .3374 .2657 .1288
2 1.0936 .7655 .5897 .4768 .3458 .2712 .1303
3 1.0407 .7364 .5695 .4642 .3388 .2667 .1292
D 1.8477 1.3020 1.0045 .8174 .5954 .4682 .2263
P 1.8022 1.2791 .9908 .8084 .5907 .4653 .2257
10 1 1.8379 1.2960 1.0005 .8146 .5938 .4672 .2260
2 2.0026 1.3892 1.0605 .8564 .6175 .4824 .2299
3 1.8579 1.3081 1.0085 .8203 .5972 .4694 .2266
D 2.6327 1.8503 1.4251 1.1584 .8425 .6619 .3193
P 2.5731 1.8206 1.4075 1.1468 .8365 .6583 .3185
15 1 2.6157 1.8402 1.4181 1.1531 .8394 .6599 .3189
2 2.8997 2.0022 1.5234 1.2273 .8819 .6874 .3259
3 1.8616 1.4325 1.1636 .8456 .6639 .3199
 85 
TABLE 4.3
Upper 5% points of T
(m = 3)
v1 30 40 50 60 80 100 200
D 1.3504 .9389 .7189 .5824 .4219 .3305 .1589
P 1.2773 .9024 .6973 .5680 .4143 .3261 .1578
5 1 1.3041 .9152 .7047 .5729 .4168 .3275 .1582
2 1.4749 1.0061 .7611 .6112 .4379 .3408 .1614
3 1.3785 .9544 .7288 .5892 .4257 .3332 .1595
D 2.2913 1.5848 1.2101 .9783 .7070 .5534 .2652
P 2.1863 1.5341 1.1805 .9590 .6970 .5474 .2638
10 1 2.1853 1.5442 1.1859 .9623 .6986 .5482 .2640
2 12.6151 1.7620 1.3219 1.0053 7499 5808 2719
32.3590 1.6221 1.2337 .9945 .7161 .5592 .2666
D 3.1977 2.2039 1.6791 1.3555 .9778 .7645 .3654
P 3.0638 2.1406 1.6427 1.3320 .9658 .7572 .3638
15 1 3.0826 2.1469 1.6447 1.3322 .9653 .7568 .3637
2 3.7372 2.5019 1.8684 1.4865 1.0514 .8116 .3773
3 ... 2.2629 1.7163 1.3811 .9920 .7736 .3677
 6 
TABLE 4.4
Upper 1% points of T
(m = 3)
V1 30 40 50 60 80 100 200
D 1.3638 .9524 .7313 .5934 .4308 .3381 .1629
P 1.3140 .9274 .7165 .5836 .4257 .3350 .1622
5 1 1.3589 .9491 .7290 .5918 .4299 .3375 .1627
2 1.4621 1.0062 .7653 .6168 .4439 .3465 .1650
31 1.3689 .9556 7335 5951 .4319 .3389 .1631
D 2.4501 1.7053 1.3070 1.0593 .7679 .6022 .2896
P 2.3732 1.6676 1.2847 1.0447 .7603 .5975 .2886
10 1 2.4357 1.6971 1.3017 1.0553 .7654 .6005 .2891
2 2.7213 1.8562 1.4033 1.1261 .8056 .6264 .2957
3 .. 1.7166 1.3143 1.0644 7709 .6041 2901
D 3.5074 2.4358 1.8643 1.5095 1.0930 .8565 .4113
P 3.4069 2.3871 1.8359 1.4910 1.0834 .8507 .4100
15 1 3.4830 2.4212 1.8560 1.5034 1.0896 .8545 .4107
2 3.9696 2.6961 2.0318 1.6266 1.1597 .8996 .4223
3 ... 1.8773 1.5186 1.0982 .8599 .4121
 7 
TABLE 4.5
Upper 5% points of T
(m = 4)
V2
V1 30 40 50 60 80 100 200
D 1.7348 1.1892 .9040 .7289 .5253 .4106 .1962
P 1.6178 1.1324 .8706 .7071 .5139 .4035 .1946
5 1 1.6745 1.1595 .8865 .7174 .5193 .4069 .1953
2 1.9498 1.3036 .9749 .7773 .5519 .4273 .2003
31 1.7819 1.2147 .9200 7399 5314 .4144 .1972
D 2.9810 2.0356 1.5441 1.2434 .8946 .6985 .3332
P 2.8126 1.9564 1.4985 1.2139 .8795 .6894 .3312
10 1 2.8785 1.9865 1.5155 1.2244 .8843 .6920 .3316
2 3.5341 2.3341 1.7301 1.3716 .9657 .7437 .3444
3 .. 2.0957 1.5816 1.2690 .9087 .7075 .3354
D 4.1858 2.8501 2.1581 1.7359 1.2471 .9729 .4632
P 3.9706 2.7510 2.1019 1.6999 1.2289 .9619 .4608
15 1 4.0413 2.7801 2.1188 1.7097 1.2335 .9648 .4613
2 5.1018 3.3493 2.4732 1.9531 1.3686 1.0506 .4827
3 . .. 2.2170 1.7760 1.2692 .9869 .4666
SS
TABLE 4.6
Upper 1% points of T
(m = 4)
 9
Pillai's approximation, were obtained from Tables of the
Incomplete Beta Distribution [28]. There are five entries
in each cell. For m = 2, these correspond to the exact
percentage points and the approximate percentage points
found from Pillai's approximation and approximations 1,
2 and 3 of this chapter. These are designated by E, P, 1,
2, and 3 respectively. For m = 3 and 4 the exact per
centage points are replaced by those given by Davis which
are designated by D. Davis' percentage points are then
used as a standard by which to compare approximations.
The omitted percentage points for approximation 3 are
those which occurred in cases where v2 < (m 1) vl + 1.
4.3 Summary
A comparison of the preceding tables reveals the
following generality for all approximations. The 5% points
are uniformly more accurate than the corresponding 1%
points, and the accuracy of both the 5% and 1% points
increased with v2, while decreasing with both vi and m.
Some unexpected results are also revealed. The
5% and 1% points of T computed from the X2 apprOXimatiOR
(approximation 1) are surprisingly close to the exact
values for m = 2 and the values given by Davis in the
cases m = 3 and 4, even for small v!2. In COmnparing this
approximation with Pillai's, the X2 appTOXimation is
 90 
generally more accurate for both the 5% anid 1% points.
Some exceptions occur in the 1% points when both vi and
V2 are large.
Although it was hoped that approximation 2 would
be relatively accurate overall, the results were rather
dis appointing This approximnation is, without exception,
less accurate than the others. It is, however, very
simple, and fairly accurate for large values of v2.
The results of approximation 3 were very satisfying.
Both the 5% and 1% points for this approximation were gener
ally more accurate than Pillai's, and markedly so for small
V2. The decrease in accuracy for increasing vl seems to
be less for approximation 3 than for the other approximations.
The fact that this approximation cannot be applied when
V2 < m 1) v1 + 1 Should not be too disturbing, since in
most practical situations we would expect v2 to be large,
relative to vi.
Although the comparison of percentage points is
only for the cases m = 2, 3 and 4 and selected values of
vl and v2, it is suspected that the relative accuracy of
the approximations would be maintained for general m, vl
and V2. It is therefore suggested that the distribution
of T be approximated by
,,,i=1
 91 
when v2 > (m 1) V1 + 1. In this case the value of v is
given by
my v 2 m)(432
Sv2(m1)vl+1' (4 '2
If the situation should arise when v2 < E 1) 1i i,
the X2 apprOXimation is suggested.
T aX A.(4.3.3)
v mvi 1
in which
a=(v21)(vlfv2m1 ) (4.3.4)
( V2m) (v2m1)(v2m3)
and
y v l2 m) (v2m 3) (4.3.5)
(v21)(V2*M2m1)
