POPULATION DENSITY ESTIIMATION USING
LINE TRANSECT SAMPLING
BY
JOHN A. ONDRASIK
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
1979
To Toni
For Her Love and Support
ACKNOWLEDGMENTS
I would like to thank my adviser, Dr. P. V. Rao, for his
guidance and assistance throughout the course of this research.
His patience and thoughtful advice during the writing of this
dissertation is sincerely appreciated. I would also like to
thank Dr. Dennis D. Wackerly for the help and encouragement
that he provided during my years at the University of Florida.
Special thanks go to my family for the moral support
they provided during the pursuit of this degree. I am espe
cially grateful to my wife, Toni, whose love and understand
ing made it possible for me to finish this project. Her
patience and sacrifices will never be forgotten.
Finally, I want to express my thanks to Mrs. Edna Larrick
for her excellent job of typing this manuscript despite the
time constraints involved.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . . iii
LIST OF TABLES ... . .... . . . . . . . .vi
ABSTRACT . . . . . . . . . . . . vii
CHAPTER
I INTRODUCTION . . . . . . . . . 1
1.1 Literature Revieu . . . . . . 1
1.2 Density Estimation Using Line Transects 4
1.3 Summary of Results . . .. . . . .9
II DENSITY ESTIMATION USING THE INVERSE
SAMPLING PROCEDURE . . . . . . . 13
2.1 Introduction . . . . . . . 13
2.2 A General Model Based on Right Angle
Distances and Transect Length . . .. 14
2.2.1 Assumptions . . . . . 15
2.2.2 Derivation of the Likelihood
Function . . . . . . 16
2.3 A Parametric Density Estimate . . 28
2.3.1 Maximum Likelihood Estimate for D 28
2.3.2 Unbiased Estimate for D . . .. 29
2.3.3 Variance of 6 . . . .31
2.3.4 Sample Size Determination Using u 32
2.4 Nonparametric Density Estimate . . . 34
2.4.1 The Nonparametric Model for
Estimating D . . . . . . 36
2.4.2 An Estimate for fy(O) . . .. 37
2.4.3 Approximations for the Mean and
Variance of (0) . . . 40
2.4.4 A Monte Carlo Study . . . 42
2.4.5 The Expected Value nnd Variance for
a Nonparamctric Estimate of D. . 46
2.4.6 Sample Size Determination Using DN 47
TABLE OF CONTENTS (Continued)
CHAPTER Page
III DENSITY ESTIMATION BASED ON A COMBINATION
OF INVERSE AND DIRECT SAMPLING .. . . .. 49
3.1 Introduction . . . . . . . 49
3.2 Gates Estimate . . . . . . . 50
3.2.1 The Mean and Variance of 6 .. ... 54
g
3.3 Expected Value of DCp ... ... .... 57
3.4 Variance of DCp . . . . . . 65
3.5 Maximum Likelihood Justification for DCP. 69
IV DENSITY ESTIMATION FOR CLUSTERED POPULATIONS .71
4.1 Introduction . . . . . . 71
4.2 Assumptions . . . . . . . . 73
4.3 General Form of the Likelihood Function .76
4.4 Estimation of D when p() and h()
Have Specific Forms . . . . . . 79
4.5 A Worked Example . . . . . . 86
BIBLIOGRAPHY .. . . . . . .. . .. . 90
BIOGRAPHICAL SKETCH . . . . . . . .92
LIST OF TABLES
TABLE Page
1 Number of animals, No, that must be sighted to
guarantee the estimate, D has coefficient of
variation, CV(Du) . . . . . .. . 34
2 Forms proposed for the function, g(y) . . .. 36
3 Results of Monte Carlo Study using g,(y) =e10v 45
4 Results of Monte Carlo Study using g2(y) = y 45
5 Results of Monte Carlo Study using g3(y) = 1y 46
6 Number of animals, No, that must be sighted
to guarantee the estimate DN has coefficient
of variation, CV(DN) . . . . . . . 48
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
POPULATION DENSITY ESTIMATION USING
LINE TRANSECT SAMPLING
By
John A. Ondrasik
December 1979
Chairman: Pejaver V. Rao
Major Department: Statistics
The use of line transect methods in estimating animal
and plant population densities has recently been receiving
increased attention in the literature. Many of the density
estimates which are currently available are based only on
the right angle distances from the sighted objects to a
randomly placed transect of known length. This type of sam
pling, wherein an observer is required to travel along a line
transect of some predetermined length, will be referred to
as the direct sampling method. In contrast, one can use an
inverse sampling plan which will allow the observer to termi
nate sampling as soon as he sights a prespecified number of
animals.
An obvious advantage of an inverse sampling plan is that
sampling is terminated as soon as the required number of
objects are sighted. A disadvantage is the possibility that
sampling may not terminate in any reasonable period of time.
Consequently, a third sampling plan, in which sampling stops
vii
as soon as either a prespecified number of objects are sighted
or a prespecified length of the transect is traversed, is of
practical interest. Such a sampling plan will be referred to
as the combined sampling method. The objective of this dis
sertation is to develop density estimation techniques suit
able for both inverse and combined sampling plans.
In Chapter II, both a parametric and a nonparametric
estimate for the population density are developed using the
inverse sampling approach. We will show that a primary
advantage of estimation using inverse sampling is the fact
that these estimates can be expressed as the product of two
independent random variables. This representation not only
enables us to obtain the expected value and variance of our
estimates easily, but also leads to a simple criterion for
sample size determination.
In Chapter III, we derive a parametric density estimate
that is suitable for the combined sampling method. This esti
mate will be shown to be asymptotically unbiased. An approx
imation to the variance of this estimate is also provided.
The density estimates developed in Chapters II and III
are based on the assumption that the sightings of animals are
independent events. In Chapter IV we relax this assumption
and develop an estimation procedure using inverse sampling
that can be applied to clustered populationsthose popula
tions composed of small groups or "clusters" of objects.
viii
CHAPTER I
INTRODUCTION
1.1 Literature Review
Our objective in this dissertation is to examine the
problem of density estimation in animal and plant populations.
The demand for new and more efficient population density esti
mates has grown quite rapidly in the past few years. Anderson
et al. (1976, p. 1) give a good assessment of the present sit
uation and provide some reasons for the renewed interest in
this subject in the following paragraph:
The need to accurately and precisely estimate the
size or density of biological populations has increased
dramatically in recent years. This has been due largely
to ecological problems created by the effects of man's
rapidly increasing population. Within the past decade,
we have witnessed numerous data gathering activities
related to the Environmental Impact Statement (lJEPA)
process or Biological Monitoring programs. Environmental
programs related to phosphate, uranium and coal mining
and the extraction of shale oil typically require esti
mates of the size or density of biological populations.
The Endangered Species Act has focused attention on the
lack of techniques to estimate population size. It now
appears that hundreds of species of plants may be pro
tected under the Act, and, therefore, we will need infor
mation on the size of some plant populations. Estimation
of the size of biological populations was a major objec
tive of the International Biological Program (IBP) (Smith
et al. 1975). Finally, we mention that the ability to
estimate population size or density is fundamental to
efficient wildlife and habitat management and many impor
tant studies in basic ecological research.
The estimation of population size has always been a very
interesting and complex problem For a recent review of the
general subject area see Seber (1973). Although many of the
methods described in Seber's book are quite useful, they are
frequently very expensive and time consuming. Estimation
methods based on capturerecapture studies would fall into
this category. A further problem with many estimation meth
ods is that they are based on models requiring very restric
tive assumptions which severely limit their use in analyzing
and interpreting the data. For these reasons and others,
line transect sampling schemes are becoming more and more
popular. This method of sampling requires an observer to
travel along a line transect that has been randomly placed
through the area containing the population under study and
to record certain measurements whenever a member of the popu
lation is sighted. There are several density estimation tech
niques available using line transect data; however, the full
potential is yet to be realized.
Density estimation through line transects is typically
practical, rapid and inexpensive for a wide variety of popu
lations. Published references to line transect studies date
back to the method used by King (See l.copold, 1933) in the
estimation of ruffed grouse populations. Since that time,
numerous papers investigating line transect models have
appeared, e.g., Webb (1942), Hayne (1949), Robinette et al.
(1954), Gates et al. (1968), Anderson and Pospahala (1970),
Sen et al. (1974), Burnham and Anderson (1976) and Crain
et al. (1978). Since it is commonly assumed by these authors
that the objects being sampled are fixed with respect to the
transect, line transect models are best suited for either
immotile populations, flushing populations (populations where
the animal being observed makes a conspicuous response upon
the approach of the observer) or slow moving populations.
Examples of such populations are:
(i) immotile birds' nests, dead deer and plants,
(ii) flushing grouse, pheasants and quail, and
(iii) slow moving desert tortoise and gila monster.
The degree to which line transect methods can be applied to
more motile populations, such as deer and hare, will depend
on the degree to which the basic assumptions are met. In any
case, one should proceed cautiously when using these models
for motile populations.
Despite the wide applicability of line transect methods,
the estimation problem has only recently begun to receive
rigorous treatment and attention from a statistical standpoint.
Gates et al. (1968) were the first to develop a density esti
mation procedure within a standard statistical framework.
After making certain assumptions with regard to the probabil
ity of sighting an animal located at a given right angle
distance from the transect, they rigorously derived a popu
lation density estimate. In addition, they were the first
authors to provide an explicit form for the approximate sam
pling variance of their density estimate.
1,
While the assumptions of Gates et al. (1968) concerning
the probability of sighting an animal did work well for the
ruffed grouse populations they were studying, it is clear
that the validity of their assumptions will be quite crucial
in establishing the validity their density estimates. If
the collected data fail to substantiate their assumptions,
large biases could occur in the estimates as seen in Robinette
et al. (1974). As a result, Sen et al. (1974) and Pollock
(1978) relaxed the assumptions of Gates et al. (1968) by
using more general forms for the sighting probability, while
Burnham and Anderson (1976) developed a nonparametric approach
as a means of providing a more robust estimation procedure.
In the following sections, we will outline the general
problem of density estimation using line transects, give
our approach to the solution of this problem and summarize
the results found in the remainder of this work.
1.2 Density Estimation Using Line Transects
The line transect method is simply a means of sampling
from some unknown population of objects that are spatially
distributed. In the context of animal or plant population
density estimation, these objects take the form of mammals,
birds, plants, nests, etc., which are distributed over a par
ticular area of interest. From this point on, our refer
ences will always be to animal populations with the under
standing that the estimation methods we describe are appli
cable to all populations which satisfy the necessary assump
tions.
In the line transect sampling procedure, a line is ran
domly placed across an area, A, that contains the unknown
population of interest. An observer follows the transect and
records one or more of the following three pieces of informa
tion for each animal sighted:
(i) The radial distance, r, from the observer to the
animal.
(ii) The right angle distance, y, from the animal to the
line transect.
(iii) The sighting angle, 8, between the line transect
and the line joining the observer to the point at
which the animal is sighted.
These measurements are illustrated in Figure 1.
Figure 1. Measurements recorded using line transect sampling.
(Z is the position of an observer when an animal
is sighted at X. XP is the line from the animal
perpendicular to the transect.)
In this work, we shall consider the problem of estimating
population density using only the right angle distances.
Because estimates depending only on right angle distances are
easy and economical to use, such estimates have become very
popular over the past several years.
Before any estimation procedure based on right angle
distances can be formulated, certain assumptions regarding
the population of interest must be made. A set of assump
tions used by several workers in the area is detailed in
Section 2.2.1. One of the key assumptions in this set is
that the probability of sighting an animal located at a right
angle distance, y, from the transect can be represented by
some nonincreasing function g(y), which satisfies the equal
ity, g(0) = 1. This function is simply a mathematical tool
for dealing with the fact that animals located closer to the
line transect will be seen more readily than animals located
further away from the transect. An alternative method of
dealing with this phenomenon is given by Anderson and
Pospahala (1970).
If g(y) is assumed to have some specific functional form
determined by some unknown parameters, then the estimate is
said to be parametric. On the other hand, if g(y) is left
unspecified except for the requirements that it is nondecreas
ing and g(0) =1, then the estimate is said to be nonparametric.
Seber (1973) has shown that any density estimate based on
right angle distances will have the form
N
s 2LoC'
where N is a random variable representing the number of
animals seen in a line transect of length Lo and c is an
estimate for c, a parameter which depends on g(y) through
the relation
c = g(y)dy.
By noting that the density is simply the number of animals
present per unit of area, it is clear that c can be inter
preted as onehalf of the effective width of the strip actu
ally covered by the observer as he moves along the transect.
Further examination of Ds also points out that estimating
the parameter c is the key to the estimation problem.
At this time, we would like to point out that the range
for the right angle distance, y, is allowed to go from 0 to
+, as seen in the integral on the right hand side of the
equation for c. In practice, since we are considering only
a finite area, A, there will most certainly be a maximum
observation distance, W, perpendicular to the transect.
However, if W is large enough so that the approximation
Sg(y)dy g(y)dy (1.1)
is reasonable, then letting y range in the interval (0,+)
will not cause any real problems. In practical terms, this
means that the probability of observing an animal located
beyond the boundary, W, should be essentially zero.
In most real life situations, W can be chosen large
enough so that the approximation given in (1.1) is valid.
Thus, in the chapters which follow, we will implicitly
assume chat relation (1.1) holds for the density estimates
that we develop.
Both parametric and nonparamecric models have been used
to derive an estimate for the parameter c, and, consequently,
for the population density. In both cases, the estimate for
c turns out to be a function of the observed right angle
distances. In the parametric case, c will simply be a func
tion of the parameters that define the function chosen for
g(y). Examples of parametric estimates are found in Gates
et al. (1968), Sen et al. (1974) and Pollock (197S).
Estimation using the nonparametric model is more compli
cated. Burnham and Anderson (1976) have shown that estimat
ing 1 is equivalent to estimating fy(0), where fy(.) is the
conditional probability density function for right angle
distance given an animal is sighted. Thus, the problem of
finding a nonparametric estimate for the population density
reduces to the problem of estimating a density function at a
given point. Unfortunately, this problem has not received
much attention in the literature. Burnham and Anderson (1976)
suggest four possible estimates for fy,(0), but the sampling
variances associated with these estimates have not been
established.
Crain et al. (1978) have also considered the problem of
estimating fy(0). They derive an estimate using a Fourier
Series expansion to approximate the conditional probability
density function fy(y). Although their procedure does not
lead to a simple estimate, they do provide an approximation
to its sampling variance.
The line transect method and the corresponding population
density estimates so far described require the observer to
travel a predetermined distance, Lo, along the transect.
This methodwillbe called the direct sampling method. An
alternative to the direct method is the inverse sampling
method, wherein sampling is terminated as soon as a speci
fied number, No, of animals are sighted. Clearly, in the
direct method, the number of animals seen is a random variable
and the total length travelled is a fixed quantity, while in
inverse sampling method, the total length travelled is the
random variable and the number of animals chat must be seen
is fixed. The main focus of this work will be to develop
density estimation techniques that are based on the inverse
sampling method. In addition, we will consider the density
estimation problem when a combination of the inverse and
direct sampling plans is used.
1.3 Summnar. of Results
In Chapter 2 we derive two estimates for the population
density, D, using an inverse sampling scheme. The set of
assumptions which justify the use of these estimates is
similar to those used by Gates et al. (1968) and several
others. The estimates have the form
D =
where No is the number of animals thac must be seen before
sampling terminates, L is a random variable representing the
length travelled on the transect and c is as previously
defined. Note the similarity of bI to Ds given in Section 1.2.
The only difference between the two estimates is that in bI
the random variables are L and E, while in s they are II and c.
However, this difference gives the inverse sampling method a
theoretical advantage over the direct sampling method. The
random variables L and E will be seen to be independent while
N and c are not. Thus, the estimate DI is the product of two
independent random variables, a fact which not only allows
us to obtain its expected value and variance easily, but also
leads to a simple criterion for sample size determination.
Both a parametric and a nonparametric estimate for the
animal population density are developed in Chapter II. In
deriving the parametric estimate, the functional form assumed
for g(y) is identical to the one used by Gates et al. (1968).
Our parametric density estimate is shown to be unbiased and
the exact variance of this estimate is also provided.
In the nonparametric case we propose an estimate for
f (0) using the method developed by Loftsgaarden and
Quesenberry (1965). We then use heuristic reasons to show
that the corresponding density estimate is asymptotically
unbiased, and derive a large sample approximation for its
variance.
The inverse sampling method does have one drawback when
there is little information available concerning the popula
tion to be studied, namely, there exists the possibility that
an observer might have to cover a very long transect to sight
No animals. To overcome this problem, we develop a parametric
density estimate in Chapter III that is based on a combina
tion of the inverse and the direct sampling procedures. In
the combined sampling scheme, sampling is terminated when
either a prespecified number, No, of animals are sighted or
when a prespecified length, Lo, has been travelled along the
transect. Thus, in combined sampling both the length trav
elled and the number of animals seen will be random variables.
In deriving the density estimate based on the combined
sampling method, we again use the functional form for g(y)
proposed by Gates et al. (1968). This estimate is shown to
be asymptotically unbiased. In addition, an approximate
variance for this density estimate is provided.
The density estimates developed in Chapters II and III
are based on the assumption that the sightings of animals
will be independent events. Gates et al. (1968) showed that
this assumption failed to hold for the animal population they
were studying. In Chapter IV we relax this assumption, and
develop an estimate based on inverse sampling that can be
12
applied to clustered populationspopulations in which the
animals aggregate into small groups or"cluscers." Since
the estimation procedure developed will require the use of
a highspeed computer, the last section of Chapter IV is
devoted to a worked example to illustrate the computations
that would be involved.
CHAPTER II
DENSITY ESTIMATION USING THE INVERSE
SAMPLING PROCEDURE
2.1 Introduction
In this chapter we shall propose estimates for animal
population density based on an inverse sampling procedure.
Unlike the direct sampling method considered by Gates et al.
(1968), the inverse sampling procedure specifies the number
of animals that must be sighted before the sampling can be
terminated. Thus, in the inverse case the number of animals
sighted will be a fixed rather than a random quantity.
A precise formulation of the inverse sampling method is as
follows:
1. Place a line at random across the area, A,
to be sampled.
2. Specify a fixed number, No, and sample along the
line transect until No animals are observed.
As one proceeds along the transect, certain measurements
will be made. These will be denoted by yl,y2,'... ,yj and
Z, where y. is the right angle distance from the ih animal
observed to the transect and k is the total distance trav
elled along the transect during the observation period.
A visual depiction of these measurements is given in Figure 2.
Figure 2. Measurements recorded using inverse sampling.
2.2 A General Model Based on Right Angle
Distances and Transect Length
The estimates for the density, D, that we will develop
are based on the right angle distances, yly2 ...' YNo and
the total distance, Z, travelled along the transect. TJo
possible approaches to the estimation of D merit consider
ation. First, recall that density is defined as the number
of animals present per unit of area, or equivalently the
rate at which animals are distributed over some specific
area. Therefore, we can write
D =
where A is the area of interest and N is the total number
of animals present in A. In the direct sampling approach
the estimation of D is most often accomplished by first
estimating N and then dividing by A. Seber (1973) shows
that any estimate of N based on direct sampling has the
form
^* NA
2LoC
where N is a random variable denoting the number of animals
seen, L, is the length of the transect and c is an estimate
of c, a parameter which depends on the probability of sight
ing an animal given its right angle distance from the tran
sect. Note that in Seber's estimate, N is random and Lo is
fixed. It follows then, that Seber's estimate for D does
not depend explicitly on A and has the form
B = N
s 2Loc
Therefore, the estimate of D is independent of the actual
size of A, a property that any reasonable estimate of D
should possess.
As an alternative, D itself can be regarded as the basic
parameter of interest and estimates for D can be derived
directly. This is the approach taken by Burnham and Anderson
(1976) and the one that we will follow in developing our
estimates.
2.2.1 Assumptions
The form of any estimate of D, the animal population
density, will depend upon the type of assumptions we can
make regarding the distribution of the animals to be censused
and the nature of the observations that will be made. The
assumptions our estimates will be based on are as follows:
Al. The animals are randomly distributed with rate or
density D over the area of interest A, i.e., the
probability of a given animal being in a particu
lar region of Area, 6A, is 6A/A.
A2. The animals are independently distributed over A,
i.e., given two disjoint regions of area, 6A1 and
6A2'
P(n1 animals are in A1 and n2 animals are in 6A2)
= P(n1 animals are in 6A1)P(n2 animals are in 65A).
A3. The probability of sighcing an animal depends
only on its distance from the transect. In addi
tion, there exists a function g(y) giving the
conditional probability of observing an animal
given its right angle distance, y, from the tran
sect. In probability notation,
g(y) = P(observing an animal I y).
A4. g(0) = 1, i.e., animals on the line are seen with
probability one.
A5. Animals are fixed, i.e.. there is no confusion
over animals moving during sampling and none are
counted cwice.
2.2.2 Derivation of the Likelihood Function
We will use the maximum likelihood procedure to obtain
an estimate for D. The joint density function we are inter
ested in is
f ,L(v .; No)
where Y = (Y ,Y2' ,. . Y ) is the vector of random variables
1 2 0
representing the right angle distances, L is the random var
iable representing the total length travelled, and No is the
specified number of animals to be seen before sampling ter
minates. Since the dependence of the joint density on No is
implicit throughout the rest of this chapter, it will be
dropped from our notation for convenience. Thus, from now
on we will denote the density as
fy,L ( ,'),
and all other expressions depending on No in this manner will
be handled accordingly.
The following two theorems will be very useful in the
derivation of the likelihood function.
Theorem 1: Let N(Z) denote the number of animals sighted in
the interval (0,O] along the transect. Then, N(Z) is a
Poisson process, and for some > 0,
n!
Note that the quantity O8 equals the expected number of
animals sighted per segment of length I.
Proof: In order to show that N(A) is a Poisson process,
we will show that the assumptions in Section 2.2.1 imply the
postulates necessary for a Poisson process given in Lindgren
(1968, p. 162).
First, consider two disjoint intervals, 1 and Z2' along
the transect and the corresponding areas, A(1) and A(Q2),
enclosed by lines perpendicular to the transect as shown
in Figure 3.
Figure 3. Two disjoint areas along the transect.
Now let N1 and N2 be random variables representing the total
number of animals that occupy A(Q1) and A(r.2), respectively.
By definition, N(QI) and N(S2) are the number of animals
sighted in A(UI) and A(r2), respectively. We know from assump
tion A2 that N1 and N2 are independent, and from assumption A3
that sighting an animal depends only on its distance from the
transect. Thus, N(C1), which depends solely on N1 and the
distances to the NI animals from the transect, is independent
of N(Q2), i.e., the number of sightings that occur in two
disjoint intervals along the transect are independent events.
Next we will show that for every >m> 0 and any h >0,
N(e)N(m) and N(C+h)N(m+h) are idcnticailly distributed.
First, note that the effective area sampled in seeing N(O)N(m)
animals and N(R.+h)N(m+h) animals is equal to A(.m) as seen
in Figure 4.
Figure 4. Effective area sampled in seeing N(Q)N(m)
animals and N(Z+h)N(m+h) animals.
Therefore, by assumptions Al, A2, and A3, and since the tran
sect is dropped at random, it follows that
P{N(Z)N(m) =j}=P{N(Z+h)N(m+h) =j), j =0,1,2,...
Next we must show that for every Z > 0, and some 9 > 0,
P{N(z) = 1 = B+o(Z), as O,
where o(Z) is a function such that
lim () 0.
Again let A(Z) be the area defined by t on the transect. Noow
define B. to be the event {N(P.)=j} and E. to be the event
J J
that there are exactly j animals in area A(Z). Then it fol
lows that
0o
P(B ) = E P(BIE )
j=l
= P(BIEj)P(Ej).
j=1
20
Under assumptions Al and A2, Pielou (1969, p. 81) has shown
that
DA(C)
P(Ej) DA()
P(E ) = e j =0,1,2... .
Also, under assumptions Al, A2 and A3, Seber (1973, Eq. (2.6))
has shown that
P(BIEl) i T
where
c = g(y)dy. (2.1)
'0
Therefore, we can write
P(B1) = 2cDe DA(Q) + 7 P(B 1Ej)P(E )
j=2
and if we show
SP(BlIE.)P(E.) = o(C)
j=2
the proof will be complete. Note that
CO M DA()
e DA()[DA(C)1J
P(B1 E.)P(E ) Z e .IDA(
j=2 j=2 J.
SDA() jDA(9) 1
= DA(.) L e'
j=2
C" DA(9.) j1
e AIDA(e)] D
<j DA() 7
j=2. (j1)!
j 2
= DA(.)I1eDA()I.
For any finite area A, A(Z) is 0(Z), that is
lim A() < K, for some K> 0.
', 
Therefore, as + 0
E P(BI E.)P(E ) = o().
j=2
and,. upon writing
0 = 2cD, (2.2)
we get, as 0,
P(B ) = e9 + o(Z).
Finally, we need to show that for every Z > 0,
E P{N() =n} =o(Z), as 0.
n>l
Note that for all n> 1, we can write
P(B ) = E P(B Ej)
j=l
= P(B IE.)P(Ej ).
j=n
Again, by using the fact that A(Z) is 0(,), it is easy to
show that
P(Bn) = o(9A), as 0,
and N(k) satisfies the four conditions necessary for a Poisson
process.
Before proceeding to the second theorem, we need to define
the following random variables. Let T. denote the random
i~
variable corresponding to the distance travelled on the
transect between sightings of the (i1)st and ith animals,
i = 1,2,... ,N. Then the total distance travelled is given by
No
L = T..
i=l
The following theorem establishes the independence of
Y and T1,T2 ... T1 for the case No = 2, and this fact enables
us to derive the joint density function, fyL(v,,).
Theorem 2. The random variables T1, T2, Y1 and Y2 are mutually
independent.
Proof: In order to establish the independence of T 2, T2,
Y1 and Y2 we will derive the joint density
fT1 T2 1 (t t2 l '2)
and show that it can be factored into four functions, each
depending on only one of the random variables of interest.
Let 1' Y,2' t1' t2' h1, h2' g1 and g2 be nonnegative
real numbers such that
t + h < I + t2'
l + g1 2
as shown in Figure 5.
TI~1
fIn
tl+t2 tl+t2+h2
Figure 5.
Areas defined by yl,y2,tl,t2',g1g2,hl, andh2.
Now let
P(h1,g81h2,g2) = P(t1 < T2 tl+hl'Y1 < Y 1 < 1+g
t2 < T2 t2 +h2'Y2 < Y2 y2 +g2).
Then we can write
P(hl,g' ,h2'g2)
fTl T Y Y (tlt2I'yl2) h lim h
2' '1 2 h. 0 hlgl2g2
gi 0
i = 1,2
provided the limit exists.
Now notice that the event whose probability we wish to
find, namely
{t < T st] +hl'I < Y1 <1 +glt2 < T2 t2 +h2 'y2
is equivalent to the intersection of the following events:
Si, the event {N(tl) = 0}
S2, the event {N(tl+hl)N(tl) = 1} and {yl
i.e., an animal is seen in area I
Y2
Yl+gl
Yl
t1 tl+hl
S, the event (N(t +t2) N(tl+h) = 0)
S the event (N(tl+t2+h2) N(tl+t2) = 11 and
{y2.'Y2 2+g}2, i.e., an animal is seen in area II.
Now, by Theorem land Assumption A3,.the events S1, S2, S3
and S4 are independent so that we can write
P(hl,g1,h2,g2) = P(S1S2S3S4)
= P(SI)P(S2)P(S3)P(S ).
We now need to find expressions for the probabilities of
Si, S2' S3 and S Since N(9.) is a Poisson process,
P(S ) = e ,
and
P(S3) = e0(ch)
However, P(S2) and P(SA) are not so easil: obtained. We will
only show how to find P(S2), since P(SL) is found in a similar
fashion. First, define S2j to be the event that there are
exactly j animals in area I. Then
P(S2) = E P(S2S2j)
j=1
= P(S2 S2j)P(S2j).
j=l
By assumptions Al and A2, the number of animals located in
area I will be distributed as a Poisson random variable with
parameter 2Dglhl, where D is the density of the animals (see
Pielou, 1969, p. 81). Note, the factor of 2 comes in since
area I can be found on both sides of the transect. Therefore,
P(S2j) =
2Dglhl j
e (2Dglh1)
S = 0,1,2,...
By assumption A3,
P(S21S21) = g(y),
for some yl
that found in Theorem 1, as gl0 and h*0,
2Dglh1 O
P(S) = 2Dglhle g(yl) .+ P(S21S2j)P(S2j)
j=2
2Dglh+o(g
= 2Dglhle g(y{)+o(glhl).
Similarly, we can show that as 2+0 and,h20,
P(Se2Dg2h2 +o(g2h2
P(S4) = 2Dg2h2e g(yn)+o(g~h2),
for some y2
for P(S1), P(S2), P(S3) and P(S4) into P(hl,gl,h2,g2)
to obtain
P(hl,g' ,h2,g2) =e
9t1 e(t2h1)
e {2Dglhle
2Dglh1
g(yl)+ C(glhl ))}
x { h2Dg2h2 + 2h2
x {2Dg2h2e g(y)+ g2h2)}.
Consequently,
P(h,g ,h2'g2) 2 0t1 0t2
lil h" = 4D e e g(yl)g(y2),
h.*0 81hlg2h2
gi0
i=1,2
which completes the proof of Theorem 2.
(2.3)
In the same manner, we can show that the independence
established in Theorem 2 will hold for any finite number of
sightings, No. In this case if T = (T, ,T2... ,TN and
Y= (Y' ,...,' ), then (2.3) becomes
No
e~t
i=1
f T,(t,v) = 2 ODNoe 1 g(y ).
i=l
Upon using equation (2.2) in fT y(tv), we get
e t NO
f ,Y(c,v) = e'e i=1 c No
i=l 1
Thus, the marginal distributions for T. and Y. are
g(yi)
g (y i)
S iand c
Ot.
f (ti) = 9e t.> 0.
Therefore, T1,T,, .. TN are independent, identically distri
buted (iid) as Exponential random variables with parameter 8,
and
No
L = T. T.
has a Gamma distribution with parameters No and 0, i.e.,
oNo No1 e
S(9ro) = NO e .>0, 8>0.
L (4f(N0)
Furthermore, L is independent of Y.
The likelihood function for the estimation of 0 and c
can now be obtained by taking the product of fL(M) and
fy(z), i.e.,
NoM N N ole0O
L(O,c;y, ) = g(yi) F(No) (2.4)
c i=1
We will now outline how one can estimate D, the animal
population density, from the likelihood function given in
(2.4). As noted earlier, D is related to 0 and c by equation
(2.2), i.e.,
D 
Thus, the maximum likelihood estimate for D would be
where 0 and c are maximum likelihood estimates of 0 and c,
respectively, obtained from (2.4). Note that the estimate
D is the ratio of two mutually independent random variables,
one depending on L alone and the other depending on Y alone.
This property will be found to be very useful when evaluating
the moments of 6.
We have now set the framework necessary for deriving
an estimate of D. In the next section we shall obtain an
estimate for D assuming that g(y) has a particular parametric
form.
2.3 A Parametric Densicy Estimate
Any estimate for D that is derived after assuming an
explicit function for g(y) will be called a parametric esti
mate. Gates et al. (1968), using direct sampling, derived an
estimate for D assuming
g(y) = e
Using this same function for g(y), we will derive the corre
sponding estimate based on inverse sampling.
2.3.1 Maximum Likelihood Estimate for D
To estimate D we need to estimate both 0 and c from the
likelihood function (2.4). In this case
g(y) = e :>0, \>0
so that
1
c = .
Substituting for c in (2.2) yields
D = _ (2.5)
Also, by substituting for c in (2.4), the likelihood function
becomes
No
\. 'i nr n"o1 Oz
L(0,A, ,(J,) = i=1e i 0 e.0, y.>0. (2.6)
The joint maximum likelihood estimates for 0 and A can
now be easily obtained. The natural logarithm of the likeli
hood function is
No
ZnL(0,A;y,Z) = NolnAA yi+N onO+(No1)ZnOZ Inlr(No).
i=l
Taking the partial derivatives with respect to 9 and X yields
anL(9,A;y,z) N
3ZnL(,X;y,j) _N NN
Setting these equal to 0 yields
^ No
and
i=l
Substituting these estimates for 0 and A in (2.5), the
maximum likelihood estimate for D is seen to be
^^ 2
No
D 2 No
2 E y.
i=l
2.3.2 Unbiased Estimate for D
The expected value of the estimate D,developed in
0 A
= E(6)E(A)
since 0 and A are independent. Using the fact that L has a
Gamma distribution with parameters No and 0, we obtain
= .No
E(A) = E(
^.J0e
(No1) i
To derive an expression for E(.), first recall that
Y . ,Y are iid with the common density
f (y) = = Ae y>0.
C
No
Therefore, E Y. is distributed as a Gamma random variable
i=l
with parameters No and \ and
E(X) = E N
1. 0
(No 1)
Independence of U and X now yields
E(D) = E(6)E(,)
2 2
(N o 1)
= D.
(No1)
An unbiased estimate for D is, therefore, given by
u 2
No
2 Lu(2.7)
2L E Y.
i=l 1
2.3.3 Variance of D
U
No
Due to the independence of L and E Y. the variance of
i=l
D can be derived directly. We have
1)
Var (D ) = Var (N0O
2L E Y.
(Ni=l ]
M1) Var I
SVar N
L E Y.
i=l
(No1) 1 2
L E Y. L E Y.
2C Y 2
i=l i=l
No
Since L and E Y. are independent it follows that
i=l 1
Va(D) (NoI)4 EL_\E~ 1 E_2 iyi_
Var(D) = E E E (2.8)
S Y E Y.,
i=1 / i=1
Deriving the Var(D ) now reduces to the problem of evaluating
1 1 1 1
the expected values for L' N. and Expres
E Y. L EOY
i=l I i=l e
sions for these quantities are easily obtained by noting that
No
L and E Y. have Gamma distributions with respective parameters
i=l
No,0 and No,A. Straightforward calculations show that for
No>2,
1 0
E() = (N0 (2.9)
E Y.
E( y' (N 1)(1)
\i=1
1 0_ (2.11)
E (No (N2) (212)
Now using (2.5), (2.9), (2.10), (2.11), and (2.12) in (2.8)
we get
4 2 A 2 9
Var(D ) 0 4 2 AF
u 4 (No )2(No )2 (Nol)4
(toL) 0A j 1 1
(No2) (1o1)
= D2 (2 (2.13)
(No2 )
provided No>2. Note that Var(D ) does not exist if Nos2.
2.3.4 Sample Size Decermination Using D
The first problem in designing a survey using the inverse
line transact method is to determine in advance the number
of animals, No, that must be sighted before sampling terminates.
One criterion for the selection of No (see Sober, 1973) is Lhe
requirement that the design must yield an estimate of the
density, D, with a prescribed coefficient of variation,
CV =
E(D)
where o^ and E(D) denote, respectively, the standard deviation
D
and the expected value for the estimate, D. As one can
see immediately, small values of CV are desirable since this
indicates that the estimate has a small standard deviation
relative to its expected value.
With the inverse sampling method, the value of No needed
to guarantee a preset value, C, for the coefficient of varia
tion of D can be calculated easily. Using (2.7) and (2.13)
u
we see that, for No>2
CV(D) (2N3)/2
CV( U) = N.
uNo2
Then, setting C= CV(D ), it is easily shown that No is the
root of the quadratic equation
C2N2 (4C2+2)No+4C2+3 =0.
Solving for No yields the two roots
S1+(+C2)l/2
No = 2+
C
Since the variance of D exists only for No>2, the required
sample size is
N0 = 2+ 1++C2 )/2
C
For example, if C= .25, then No =35. Table 1 gives values of
No corresponding to coefficients of variation ranging from
.1 to .5.
Table I. Number of animals, No, that must be sighted to
guarantee the estimate, D has coefficient of
variation, CV(D ).
U
CV(Du) No
.50 11
.40 15
.30 25
.25 35
.20 53
.15 92
.10 203
2.4 Nonparametric Density Estimate
In this section we will consider a nonparametric estimate
for the population density, D, using inverse sampling. In
contrast to the parametric approach used in Section 2.3,
the nonparametric approach leaves the function g(y), which
represents the probability of observing an animal given its
right angle distance, unspecified.
In Section 2.2.2 we showed that an estimate for D is
given by
D 
2c
where I and c Jare tLhc stimates for 0, Lhe cxpcctcd number of
sightings per unit length of the transect, and c defined as
c = g(y)dy.
*O
If g(y) is completely specified, except for some parameters,
then the problem of estimating D reduces to the problem of
estimating 6 and the parameters in g(y). In Section 2.3 we
considered the specific case
g(y) = e
A drawback to this approach, where we specify a functional
form for g(y), is that the function chosen must take into
account the inherent detection difficulties that are present
when a particular animal species is being sampled. If one
examines the various forms that have been suggested for g(y),
one quickly becomes aware of the problem of finding a form
that is flexible enough to accommodate the many possibilities
which exist. Some of the functions that have been proposed
for g(y) are presented in Table 2. As seen in the table, the
suggestions for g(y) represent a number of different shapes
in an effort to reflect the nature of the animal being sampled
and the type of ground cover being searched.
Because of the problems that can arise in choosing a
function for g(y), Burnham and Anderson (1976) considered
a nonparametric approach as a means of avoiding the need for
the specification of g(y). Leaving g(y) unspecified will
allow the estimation procedure to depend on the observations
that are actually miadc, not on any panrticulnr model. Thus,
a nonparametric model might provide a more robust estimation
method, that is, an estimation method that could be applied
to a much wider class of animal species.
Table 2. Forms proposed for the function, g(v).
Function
e
Author
, A>0
Gates et al. (1968)
g(y) =
g(y) =
a
1 
0
0<: y < w
Eberhardt (1968)
'y>
1 O
0 >w
Seber (1973)
S..a P..
BX el
g(y) = e (c dx,
F(ci
y
B>0, a>0
Sen et al. (1974)
, p>O, \>0
Pollock (1978)
2.4.1 The [Jonparametric Hodel for Estimating D
Consider the estimate for D developed in Section 2.2.2,
that is
D 2a
As noted earlier, if
g(y) = e 
then
c= 1
A
and our estimate for D is
2
g(y) =
e
g(y)
Now, if g(y) is left unspecified, then an estimate for 1 may
be obtained along the same lines Burnham and Anderson (1976)
used in the case of direct sampling. By assumption A4,
f (0) = gO) 1
Y c c
1
Hence, f equals the value of the fy(') evaluated at y=0,
where fy() is the probability density function for the right
angle distance, Y, given an animal is seen. The problem of
1
finding a nonparametric estimate for , therefore, reduces
to the problem of finding an estimate, y(0), for fy(0).
An estimate for D will then be given by
D = (2.14)
where 0 may be taken as the maximum likelihood estimate derived
in Section 2.3.1. That is,
S(N 1)
L
where we have replaced No by Nol to remove the bias.
2.4.2 An Estimate for f (0)
 
Burnham and Anderson (1976) suggested four possible
methods for estimating fy(0), but we are not aware of any
work which investigates the theoretical properties of any
of these estimates. Loftsgaarden and Quesenberry (1965) con
sidered a density function estimate based on the observation
that
hat F (x+h) Fy(xh)
fy(x) = lim 2h
h0 h
where F y() is the cumulative distribution function. For the
purpose of estimating fy(0), their estimate takes the form
Ff(0) = {(O (' + ) (2.15)
where [ITfo + 11] is the value of ,Ton + 1 rounded off to the
nearest integer and Y is the j order statistic of the
sample ylY2',. n
Loftsgaarden and Quesenberry (1965) showed that f (0)
as given in (2.15) is a consistent estimate, provided fy()
is a positive and continuous probability density function.
One nice property of y(0) is that it can be easily calcu
lated from the data. However, evaluation of the moments of
this estimate does present some problems. In fact, the mean
and the variance may not even exist in some cases. But,
whenever ['TJo + 1] 3, i.e., whenever (Jo,,, the variance of
fy(0) is finite as shown in the following theorem.
Theorem 3. Let Y 1 Y2' .. n be a set of independent, iden
tically distributed random variables, representing the right
angle distances, with continuous probability density function
(p.d.f.)
fy(y) = c v>O
Also, let Y( be the rh order statistic. Then
(r)r)
Efor e y i er + h
for every integer r, such that 3 r n.
Proof: The density function for Y(r) is
hr(y) = n FrI y)[lF(y) nrfy)
where
y
F (y) = fy(t)dt.
Therefore,
E) = n1 r (y)[1Fy(y) nrdF,
Since g(y) represents a probability, g(y)sl and
F(y) g(t) dt < y
F(y) c C
Therefore,
nr
E 2 Fr(y)[lFy(y)]nrdFy),
(rY c 0 y
(nl\
Sr r (r2)r(nr+l)
c r(n1)
which completes the proof.
Simple asymptotic approximations for the mean and variance
of y(0) which work well for several densities given in
Table 2 can be developed using the first order terms in a
formal Taylor series expansion of Jy(0). The basic ideas
involved in the derivation of these approximations are pre
sented in the following section.
2.4.3 Approximations for the Mean and Variance of f,,(0)
1
Let F() and F (*) denote the cumulative distribution
function and its inverse for the random variable Y, the right
angle distance. Also, let
r = ['N + 1],
U = F(Y ),
r (r) '
and
+(U ) =
(rl)F (U )
r
th
where Y(r) is the r order statistic in a random sample of
size N from F(). Then proceeding as in Lindgren (1968,
p. 409) it is easy to see that
f,,(0) = ,(U ),
E(U)  Pr'
and
Pr(1Pr)
Var(Ur) N+2 (2.16)
Assuming that q() is continuous and differentiable once at
Pr, the first order terms in the Taylor series expansion of
F(*) at Pr yields the approximation
.(U .) t ,(p ) + (U p ) (u) (2.17)
r du u=P (2.17)
Taking expectaLions on both sides of (2.17) yields
EI t.,(U )} J l (pr),
r 
and substicucing for r, pr and g() fields
E{ (O0) ) 1
S F 1N+1
/ N /+1
Taking the limit as N tends to infinity and noting that
F (0) is 0 and u = F(y) yields
lim E{f ,(0) lim
N co Y I co
] =
u=0
SdF( o) ( ).
dy y=0
(2.18)
Thus for large N, fy(0) is approximately unbiased.
An appro::imation for the variance of 'y(0) is found
in a similar fashion. Using (2.17) we get
Var{((Ur)) d, upu r 2Var(Ur).
 u=pr
Evaluating the derivative yields
d., (u) 1
du u=pr (rl){F (pr f
so that
Varf.+(Ur)} r 1 1 
r (rl)F (r Pr)
Var (U ).
r
How, using (2.16) and then making the appropriate substitu
tions for r, pr and +*(*), we get
1 (,/ +i) (14 ,'N)
Var(f (0)) >. 
1 2
TIN+y IY N+1 I
12
Therefore, as [1u we have
lim ,'T(Var .',(0)} = f (O)
so that an approximation for the variance, when N is large,
is given by
f2(0)
Var(,,(0)} (2.19)
As stated earlier, the expressions obtained for the
expected value and variance of f,,(0) are only approximations.
Their adequacy for practical purposes may be evaluated by a
Monte Carlo study involving various specific forms for the
p.d.f., f,(). In the next section we will look at the results
of just this kind of simulation study.
2.4.4 A Monte Carlo Study
A Monte Carlo study was used to examine the approximations
for E{ F,(0)) and Var(y,(O)} presented in Section 2.4.3. Three
possible shapes for Cy() were used in the study. Since the
shape of fy(.) depends solely on the choice of g(y), the
functions
g1(y) = e10y, y>O
g2(y) = 1y O
and
g3(y) = ly2, 0
were chosen. The function gl() was first proposed by Gates
et al. (1968), while Eberhardt (1968) suggested both g2()
and g3(*). The different shapes that these three functions
represent are depicted in Figure 6.
1
82
g(y)
1 y
Figure 6. Three forms for the function g(y).
For each value of n= 25, 35, 45, 65, 80 and 100, two
thousand random samples of size n were selected from each of
the three populations defined by gi('), i= 1,2,3. These
samples were obtained by first generating observations from
a uniform distribution defined on the interval [0,1] and then
transforming these values using the appropriate density fy(*).
The UNIFORM function described in Barr et al. (1976) was used
to generate the samples from the uniform distribution. For
each set of 2000 samples, empirical estimates were calculated
for the expected value, e, the percent bias, Be, and the
standard deviation, ae, of fy(0) given in equation (2.15)
as follows:
Let fiY(0) denote the estimate from the ih sample,
i= 1,2,...,2000. Then
12000
e 1 i (0)
qe = 0 fiY(0)
i=Li
B = 00( fy(O)
e f Y(0)
and
1 2000 ( 21/2
ae i= 9 1 (f y(0) e)=l
All of the necessary computing was performed under release
76.6B of SAS (see Barr et al., 1976) at the Northeast
Regional Data Center located at the University of Florida.
The results of the study, along with the approximate
standard deviations,
fy(0)
T 
are presented in Tables 3, 4, and 5. As can be seen from the
tables, the estimate of fy(0) has a negative bias for most
samples, generally of a magnitude less than 10% of the true value.
The ratio of OT/oe is also within 10% of one for almost
all samples considered. This is even true for the smaller
sample sizes, n<45. Also, when considering the smaller sample
sizes, the ratio was for the most part greater than one.
Based on the results of this simulation, we feel that, in
practice, the approximations obtained for the expected value
and variance of fy(0) would perform adequately.
Table 3. Results of Monte Carlo Study using gl(y) = e
Sample a
Size e B oT o T
e e Te 
e
25 9.05 9.5 4.47 4.55 .98
35 9.08 9.2 4.11 4.12 1.00
45 8.87 11.3 3.86 3.65 1.06
65 9.48 5.2 3.52 3.65 .96
80 9.49 5.1 3.35 3.56 .94
100 9.48 5.2 3.16 3.13 1.01
For gl(y) the theoretical mean is 10.
Table 4. Results of Monte Carlo Study using g2(y) = ly.
Sample B T aT
Size e e T e 
e
25 1.88 6.0 .894 .850 1.05
35 1.88 6.0 .822 .801 1.03
45 1.83 8.5 .772 .674 1.15
65 1.96 2.0 .704 .722 .98
80 1.92 4.0 .669 .647 1.03
100 1.93 3.5 .632 .615 1.03
For g2(y) the theoretical mean is 2.
Table 5. Results of Monte Carlo Study using g3(y)= 1y2
o
Sample B T
B o a 
Size e e T e
e
25 1.47 2.0 .671 .625 1.07
35 1.48 1.3 .616 .594 1.04
45 1.44 4.0 .579 .559 1.04
65 1.51 .7 .528 .536 .99
80 1.49 .7 .502 .506 .99
100 1.50 0.0 .474 .477 .99
For g3(y) the theoretical mean is 1.5.
2.4.5 The Expected Value and Variance for
a Nonparametric Estimate of D
Now that we have decided upon an estimate for f (0), the
problem of estimating D is straightforward. Substituting
the estimate, fy(0), defined in Section 2.4.2 into expression
(2.14) a nonparametric estimate for D is
(N.1)
D^ = Nol) (2.20)
2L/N '1 +11)
Expressions for the expected value and variance of 5N
are easily obtained. Since L and Y,Y ...'. Yo are indepen
dent, we can write
1 ^
E(DN) = ; E(0)E{fy,(0)},
and (see Goodman, 1960, Eq. (2))
1 ar(D)
Var(N) = 4 (6)Varf (0)}+E'lf,(O))Var(6)+Var(6)Var(@'(0)],
where
^ N 1
L '
and
fy(0) = 1
/Y([/No+l])
Then, upon substituting the appropriate expressions for the
moments of 6 and fy(0) into the above equations, we get
E(DN) D, (2.21)
and
2 (/N +1)
ar(N (No+2) (2.22)
2.4.6 Sample Size Determination Using DN
We can now determine the approximate value of No that
is needed to guarantee some preset value for the coefficient
of variation of DN, CV(DN). These values for No can then
be compared to the corresponding values for No (see Table 1)
that are needed to ensure the same coefficient of variation
with the parametric estimate, D Using (2.21) and (2.22),
we see that an approximation for the coefficient of variation
of DN is
CV(6N) 1(1/2
S(No+2)172
and by setting C=CV(DN), one can easily show that /No is the
root of the quadratic equation
C2No /No + 2C21 = 0.
Solving for .''o, yields the two roots
.. t(l4C2 (2C 1))i/
fo= 2
2C
and since
(14C2(2C21))1 2 >1
whenever
? 1
C <
2'
the required sample size for values of C .5 is
Sj1+(14C2 (2C ))1/22
2C2
For example, if C= .25, then No = 284. Table 6 gives values
for No corresponding to coefficients of variation ranging
from .2 to .5.
Table 6. Number of animals, No, that must be sighted
to guarantee the estimate DN has coefficient
of variation, CV(DQI).
CV(DN) No
.50 20
.40 48
.30 142
.25 284
.20 671
CHAPTER III
DENSITY ESTIMATION BASED ON A COMBINATION
OF INVERSE AND DIRECT SAMPLING
3.1 Introduction
When sampling a population by means of line transects,
it is important to keep in mind that the transect length
that can be covered by an observer will be finite. This
poses a problem for the inverse sampling plan since there
will exist the possibility of not seeing the specified number
of animals within the entire length of the transect. There
fore, it seems reasonable to develop a sampling scheme that
would employ a rule, which allows one to stop when either a
specified number, No, of animals are seen or a fixed distance,
Lo, has been travelled on the transect. In this chapter we
will consider a sampling plan which combines the inverse
sampling procedure discussed in Chapter II and the direct
sampling procedure of Gates et al. (1968).
More precisely, we will define the combined sampling
method as follows:
1. Place a line at random across the area, A, to be
sampled
2. Specify a fixed number of animals, No>2, and a
fixed transect length, Lo, and then continue sampling
along the transect until either N animals are seen
or a distance, Lo, has been travelled.
49
Since the above method merely incorporates the individual
stopping rules from the inverse and direct sampling methods,
it seems reasonable to use the estimate
f D if H = INo
DCP = ^u (3.1)
D if N < No,
where N is a random variable corresponding to the actual num
ber of animals sighted using combined sampling, D is the
inverse sampling estimator given in (2.7) and D is an esti
g
mator appropriate for the direct sampling case. In other
words, the combined sampling procedure uses the inverse sam
pling estimate if sampling terminates after No animals are
seen and the direct sampling estimate if sampling terminates
after travelling a distance Lo. In Section 3.5 we will also
show that DCp has a maximum likelihood justification.
Before proceeding to derive the mean and variance for
DCP, we need an estimate appropriate for the direct sampling
case.
3.2 Gates Estimate
Based on the direct sampling approach and assuming
g(y) =e y, x 0,
Gates ce al. (1968) developed the estimate
0 n = 0,1
d 2(3.2)
D dn (n 1) n > 2 (
n
2L : ,
i=1
where Lo is the fixed length of the transect, n is an observed
value for the random variable Nd, the number of animals seen
using direct sampling, and yi is an observed value for the
random variable Yi, i =1,2,...,n, the right angle distance
to the ith animal seen. In what follows, we shall show that
the variance of Dd is not finite. First, we need a result
concerning the joint density of the Yi, i =1,2,...,Nd, condi
tional on Nd.
Theorem 3. Under the assumptions stated in Section 2.2.1,
conditional on Nd=n>0, the random variables Y1,Y 2...,YNd
are independently, identically distributed with common density
fy(y) = Ae y y>0, A>0.
Consequently, conditional on Nd = n>0, the random variable
Nd
E Y. has a Gamma distribution with parameters n and A.
i=l
Proof: We want to show that for yi >0, i= i,2,...,Nd,
n
A I y.
i=l
fY 1 2 (YI'' Y2 'YN INd=n) =A ne
l..YN .... YNd "
Recall that in the direct sampling procedure, the total
length travelled, Lo, is fixed, and define L to be the random
variable representing the total length travelled on the tran
sect when the nth animal is sighted. Then the events {Nd=n}
and {Ln Lo
fY1Y Y.d(yl''2.... ( I'Ndld=n)
1 2 L1 d
n
f', 1Y 2 Y ( yl ...1 nI  Lo.
Now by Theorem 2, Y, Y .Y L and L are mutually
12''' n n n+l
independent, and
g(yi)
f (Yi) g
Consequently,
n
fY Y .Y *(y1 y2 N LNd=n) = H f (yi
1 2 d i=1 l
Sn
= n R g(yi).
c 1=1
n e
i=l
which completes the proof.
It is now easy to show that Var (Dd) does not exist.
N d
From Theorem 3, conditional on 1d=n>0, Z Y. has a Gamma
i=l
distribution with parameters n and X. Thus, using (2.12)
and (3.2)
0 = 0, 1
.:(iilN d=n) = T 2 2
2 n 2
Lo (n2)
Also, since Nd is the number of sightings inatransect of
length Lo, it follows from Theorem 2 that Nd has a Poisson
distribution with parameter OL. Thus
E(D2) = E E(D2Nd)
A2 n2 (nl) e 1(0L)n
Sn2 (n2) n1
4Lo
= +0 ,
showing that the variance for the estimate Dd defined in
(3.2) is infinite. In fact as long as
P(Nd= 2) > 0,
the variance of Dd cannot be finite.
The problem of infinite variance for Dd can be overcome
by replacing Dd with D where
f0 if n=0,l,2
D =" (3. 3)
g n(n1) if n>3
2L E Y
i=l
Note that the estimate, D differs from Dd only when n=2.
Since any estimate of the density based on only 2 sightings
should be effectively 0, the above modification does not
seem to be unreasonable. We will now proceed to derive
expressions for the mean and variance of D which are needed
in the sequel.
3.2.1 The Mean and Variance of D
g
A
We will first examine E(D ). Recall from Theorem 3, that
id
conditional on 1d=n, n>0, Z Y. has a Gamma distribution with
i=l
parameters n and A. Thus
Nd, d = n
i=l
and
0 n=0,l,2
E(Dg t l=n) =nA
1, nt3
Now since Ud is distributed as a Poisson random variable with
parameter OLo, it follows that
E(Dg) = E dE(Dg INd)
C OL n
ne O(OLo)
2L n!
0 n=3
O= {le (I+OLL o)}
2(
Substituting the left hand side of (2.5) for in the above
yields
E(D ) = )D lc (1+0Lo)),
and after wriLing In = OL.o, the expected number of sightings in
a transect of length Lo, we get
E(D ) = D{1o (l+u) (3.4)
g
Thus,D is not strictly unbiased, but the bias arises because
g
there is a positive probability of obtaining samples of size
1 or 2. However, even for moderate values of v, the bias in
D will be small since e (1+p) tends to zero exponentially
g
fast. For example, if p = 10, the relative bias is only .05%.
Next we will look at Var (Dg). Again since, conditional
Nd g
on Nd=n, n>0, E Y. is distributed as a Gamma random variable,
i=l
with parameters n and A, we know that
1 2
E 1 =n X n>2,
N 2I Nd= (nl)(n2)' n>2,
E Y i
i=l 1
and
2 0 if n=0,1,2
E(D INd=n) = 2 (nl) 2
4 n (n1) if n>3
Therefore,
E(D ) = ENdE(Dg Nd)
2 0 2 n
 n (nl) e (3 5)
z. (n2) nl
n=3
4Lo
and we can write
2 0 2 n ln
Var(D ) = n (n2) ne D e +)}2 (3.6)
4Lo n=3
An approximation to Var(D ) valid for large values of u
may be derived in a manner analogous to the method used by
Gates et al. (1968). After writing
2
n (n1) 2 4
(n = +n+2+
it is easy to see that for n,3
2
n2 n+2 n (n1) 2
(n2)n+6
Thus, lower and
and
upper bounds for E(D ) are
g
.22 e n
LB =  Z (n 2+n+2)e  E(Db)
ALo n=3 n
B 2 2 uj n ^
UB = =Z n n+n+6)eE
4Lo n=3 n
Now
2 m ui n
UBLB = n
Lo n=3
2 
= 7 (1e e
2 u
 o).
Upon using the relationships
D = and = L
D = and I = 0Lo
we get
4D'
UBLB =
ji
which tends to 0 as uo
for E(Dg) is
8
(1e llie
2 I)
ue ^
Thus, a reasonable approximation
L^ UB+LB
SE(D
g = 2
\
4Lo n=3
en
2 e n
(n +n+4) 
n!
1)
 (u 2+22u+4e 11(4+6u+5ju )
U
(3.7)
From (3.7) an approximation for Var (D ) is
Var(D ) =D2{1+ 4 e ((5+ + )}D {l2e (l+l+e2 (1+p)2 }
2 2 4 ' 6 4 2 2
=D { + e (32u ++ e (1+p) }
Now, as p increases, the terms involving eP and e2p will
2 4
tend to 0 much faster than +, so that for large u, we
have the approximation
Var(D ) D2 (+ ). (3.8)
We are now in a position to derive the mean and variance
of Dp
3.3 Expected Value of DCp
Recall that in the combined sampling scheme both N, the
number of animals seen, and L, the distance travelled before
termination of sampling are random variables. Thus, the
expected value of DCp can be found directly using
E(Dp) = ENE(DCPIN).
However, before proceeding along these lines it will be help
ful to have the following theorems.
Theorem 4. Let N be the random variable representing the
number of animals seen using the combined sampling method.
Then under the assumptions stated in Section 2.2.1,
58
f n
e n=0, ,...,N 1
P(N=n)=
om n
e 1I
S n=I]o
=Uo
where u=0Lo is the expected number of animals sighted along
a transect of length Lo.
Proof: For n
event {exactly n sightings occur in (0,Lo1). By Theorem 1,
the number of sightings in (0,Lo] is Poisson with parameter
OLo. Hence,
P(N=n) = enu n=0 ....... Io1.
The case N=No follows since the event (N=N.J is equivalent
to the event (at least No sightings occur in (0,Lo]}.
The following three theorems establish some useful
relationships among the random variables I], LN andY ,Y2 .. YN
where 11 is as defined in Theorem 4, LI represents the total
th
length travelled on the transect when the th animal is
sighted and Yi, i=1,2,. ..,, represents the right angle
S th
distance to the i animal seen.
Theorem 5. Under the assumptions stated in Section 2.2.1,
the conditional p.d.f. of LN given N=n>0, is
n1
nn
n ni
Lo
fLN (N=n) =
fLeN ( ) le0 n=N
F(No) P(N=N)
where
Lo NNoNo1e 0.
P(N=No) = e dt.
'F(No)
Proof: First we will consider the case when n
Theorem 1, seeing n
to observing n Poisson events in the interval (O,Lo]. There
fore (see Bhat, 1972, p. 129), the joint density of
L1,L2 ....,LN conditional on the occurrence of N=n Poisson
events in (0,L]o is
L1 L2 ..LN( 1 . JN=n) = nn Os
and the marginal density of LN conditional on N=n is
nl
fLN(IN=n) = n! , 0
Next, we will consider the case where N=No. Define T.
to be the random variable corresponding to the distance
travelled on the transect between the (il)st and ith sight
ing. Then the Noth observation is made at
No
L = E T..
N0 i=1
60
Now in the combined sampling approach, we will see N=Noani
mals, if and only if che distance
N
L = T. L.
N0 i=l I
Therefore, if O<sLo, then
P(L tNJ=N0o)
P ( [L.= 1,
P(LIjI N=to) = P(.=)
P(i=No)
No
P( T. .)
i=l 1
P(N=No)
Now, since the sightings are Poisson events by Theorem i,
the random variable
n0
L = E T.
o i=l
has a Gamma distribution with parameters No and 0. Thus,
0[ No N I 0?.
f (IjN=N) = o[ e ___ I 0
Now, by Theorem 4, we have
P(N=[) = e0L (OLo)
j=0o j
OLz ONo No1
0 z dzo
SOe 0 NO )(o1
Sr(No) de.
Substituting P(N=No) into (3.9) above completes the proof.
Theorem 6. Under the assumptions stated in Section 2.2.1,
conditional on N=n>0, the random variables Y1,Y ... N and
LN are independent.
Proof: First consider the case N=No. Let 20 and y.i0
for i=l,2,...,N. We want to show that
P(Yi yi, LNt, i = 1,2,...,NIN=No)
= P(Y iYi, i=1,2,...,No)P(LN ]N=No).
Note that the event {N=NJ is equivalent to the event {LNo Lo},
so that we can write
P(YiYi, LN : i = 1,...,NIN=No)
= P(Y iyi, LNos, i = 1,2,...,No I=No)
= P(Y iYi, LNosz, i = 1,2,... ,NoLN sLo)
P(Yi Yi, LNo : LNoLo, i=1,2,. .. ,N )
P(LNo : Lo)
N0
Now by Theorem 2,
Consequently,
Y1Y2,...,YNo and LNo
are independent.
P(Y iYi, LNO < LN, Lo, i=1,2,...,NO)
P(LNo L0)
P(Y
P(LNo Lo)
= P(Y
= P(Y y. i=1,2,...,No)P(LNo IN=No0).
1 1 0.
Now consider the case l=n<1o, and let 9. and v. be defined
as before. Also, define XN to be the actual length travelled
to see N animals when the combined sampling method is used,
that is,
= Lo '
L No N= N< o0
L o, N=No
Then for n
are equivalent. Thus,
P(Yiy L s., i = 1,2, .... ,NN=n
= P(Y y L i = 1,2,...,nl:n=Lo)
1i n n
= P(' ., L t, i = 1,2,...,n L Lo
P(Y L , L Lo
S n n n+l)
n nn++
Again by Theorem 2, for N=n, YL 2" n and Ln Ln+
are independent, so that
P(Y.i
1 i n n n+l.
P(Y iYi, i=1,2 .... ,n)P(Ln Ln< Lo.Ln+l)
P(L nLo
= P(Y.i .Y i=l,2,...,n)P(L <. N=n
Theorem 7. Under the assumptions stated in Section 2.2.1,
conditional on [t=n>0, the random variables Y1,Y2,..' are
independently, identically distributed with common density
63
fy(y) = Ae y>0, A>O.
Consequently, conditional on N=n>0, the random variable
N
Y. has a Gamma distribution with parameters n and X.
i=l
Proof: The case N=n
by noting that the random variables N and Nd are equivalent
when 0
o
is equivalent to the event {LNo
fYY2 .... YN (Y'Y2' ... 'YN IN=No)
1 f ,Y2 ..., No (Y Y21 . 'YNo ILNo Lo )
Now by Theorem 2, Y ,Y2 .... YNo and LNo are mutually inde
pendent and
g(Yi)
fY(Yi) c
Consequently,
No
fYY2"'YNo (yl'y2' ..YN IN=N) H f (yi)
12*N. i=l i
1 No
= I g(yi),
c i=l
and substituting
AYi
g(yi) = e
completes the proof.
We are now ready to determine the expected value of
DCp given N=n. For n = 0,1,2,
E(DCPIN=n) = E(D IN=n) = 0. (3.10)
g
Next consider the values 3~n,No. Recall from Theorem 7
that, conditional on N=n>0, Y. has a Gamma distribution
i=l 
with parameters n and A. Then using expressions (3.1) and
(3.3) it follows that
E(DCpIN=n) = E(6 lN=n)
= E N(Nrl) il=n
2L.E Y.
n.\ 2 (3.11)
2L0
Finally for N=No it follows from Theorem 5, Theorem 6,
Theorem 7, and expressions (3.1) and (3.3) that
E(DcpI r=Nro) = E(Du J=No)
iE N_1 ____ E '1 [ "0
E Y L T
L 0 1 02 
2i'(N=No ) o e *
Then using the transformation 9. = we get
e\0 I, e (OL, )
E(D pN=No) = 2P0(N=N) (3.12)
CP 2P (N nl A12
n=N'. 1
We can now evaluate the expected value of DCp. Using
Theorem 4 and expressions (2.5), (3.10), (3.11), and (3.12),
we find that
E(DCP) = ENE(DcPIN)
n E L P (N=n) + E nl
n3 n=Nol
= Dfle (I+p)]. (3.13)
where i =0Lo.
Thus DCp is a biased estimate for the density. Note
that the bias here is equal to the bias for the modified
estimate, D in direct sampling. This is as expected since
in the combined sampling procedure, we are simply choosing
the estimate that corresponds to the reason for terminating
sampling. If we stop sampling after seeing the Noth animal,
then the inverse sampling estimate is used, and, likewise,
if sampling stops after travelling the distance Lo, then the
direct sampling estimate is used.
3.4 Variance of Dp
An expression for the variance of DCp can be found
directly using the formula
2 2
Var(DCp) = E(DCI) {E(D ,)}2. (3.14)
In the preceding section we derived E(Dcp) so that our
"2
problem reduces to evaluating E(DCp). Proceeding along
the same lines as in Section (3.3), we quickly find
2
E(DCPIN=n, n=0,1,2) =
E(D
CP N[=n, 2
(3.15)
S1 N=n, 2
*^
9 2
n' (nL).
4Lo(n2)
E(DCpIrJ=No) = E
CP
(3.16)
N=N j
 E IN=No E
LL
N2
1 '(N) 1) [
S4(No2) 2P (N=No)
2 2''(Nol)"
2 2
4(N02) P(N=f0)
, (N 2 I.) n=
4(No2) P(N=N0) n=N
(N, 1) 20 N 2. 3e 9.
F (No)
0 42,No3 0.
e d.
OL LO) n
eL (oLo)
n9 .0(3.17)
< 1
Then, using Theorem 4, expressions (2.5), (3.15), (3.L6), and
(3.17) and letting u=0Lo, it follows that
(2 2
E(IC) EN CP N)
2 o1
4L n=3
, ii n 2 22 / 2 2
I L(n 1 ) c A I 
(n2) nN 4 No2
n=No2
 n2 1 n2 ,u 2 2 2
,102 n u e A2 0 0
4 (n2) (n2)! ' 2
n=3 O
m 1 n
e nI
n!
n=No2
and,
0 l I 1
II
S11="No
N3
2 3n p 2 m e n
= D2 (n+2) e+ (No E e
nnN02 (3.18)
n=l n n1 n=No2 n!
An expression for the variance of DCp is now evident.
Using (3.13), (3.14), and (3.18), we get
S 2 n+2) 3 n2  / 2 co  n
Var(DCP)=D2 L (n 2)L N " [ e l+u)]2
CP n=1 n=No2 n
(3.19)
where p=9Lo.
Note that,
S rN_1 2 D2
lim Var(Dcp) = L 2 D
L 0o
and
lim Var(D p) =D e e (1+u)]
CP n n!
o400 n=1
After some simple algebraic manipulations and using the
relationship D= one can easily show that the limit as
Lof and the limit as Noo are equal to the Var(D ) given
in (2.13) and the Var(D ) given in (3.6), respectively.
These limiting values are as expected, since letting Lo0
in the combined sampling approach is equivalent to using
inverse sampling, while letting No*o is equivalent co using
direct sampling.
We will now show that Var(DCP) can be expressed as a
function of both Var(6u) and Var(b ) given in (2.13) and
(3.6), respectively. Writing the equation in this form will
then lead directly to an approximation for Var(DCp).
68
First note that (3.19) can be rewritten as
Var(Dcp) No3 (n+2) nn u e 2 1 , n
CP (n+2) e IN11 i e n u 2
mn= n n ++ 7 r lle (l+u)]
D n= n=No12
(3.20)
Adding and subtracting the terms
n=N
n=N. 2
m n u
IM (n+2) u e
n=N 2 and
n=N.2 n n!
u n
e to the right hand side of (3.20) fields
to the right hand side of (3.20) yields
n]
Var(Dcp) o n 11
CP_ = (n+2) e u 2
D = I e n +u)
D n=1
e n 
^n I ^I.2/'
co L n 0)
+ e z
n=No2 n=N,2
c n
e uI 2N,3 +
n=No2 (No2)
n u
(n+2) u e
n nT
c n 
Z (n+2) I e
n=l n
O 2 n u
2 2 u e
1le (l+u)] 2
n=No2 n n
Now, after multiplying both sides of the previous expression
2 0.\
by D using the relationship D= and substituting the
expressions for Var(D ) and Var(D ) given in (2.13) and (3.6),
respectively, we see that
11 n
Var(Dp) = Var( ) +Var(D) D 2
C n=No2 n u n=No2n n
(3.21)
Therefore, an approximation for Var(DCP) can be simply ob
tained by using the approximation for Var(D ) given in (3.8).
6
CO
+ Z
n=No,2
3.5 Maximum Likelihood Justification for Dp
In Section 3.1 we stated that the estimate
D g N
D =
DU N=No
could be justified using the maximum likelihood procedure.
To show this, we first need the joint density function for
YIY2 ... YN LN and N, i.e., fY,LN N(X,S,n).
By Theorem 6, fY,LNN() N can be written as
fY,LNN(y, ,n) =fy (yiN=n)fLN(JN=n)P(N=n).
The functional form for fY,L NN() is now evident. Using
Theorems 4 and 5 and recalling from Theorem 7 that
n
A E Y.
fy(yxN=n) = ne i=l A>O,
we obtain,
/ n
A E Y.
ne i=l n1 OLo n
nZ e (OLo)
Lo n, N=n
YLN, N(,,n) =
No
N0
A Y.
AN ei1i N ,No1 A0
(No )I N=No. (3.22)
As shown in Section 2.3. .themaximum likelihood estimate
for D is given by
OX
D 2
where 0 and \ are maximum likelihood estimates for 0 and ,\
respectively. Finding maximum likelihood estimates for
6 and A is now straightforward. Taking the natural logarithm
of the likelihood function and setting the partial derivatives
with respect to 6 and equal to 0 yields, for n>0,
N=n
96 =
and
Y N
Z Y.
,
I N=l, . N .
Thus, a maximum likelihood estimate for D using combined
sampling would be
9
N2
E'l
i=l 1
N=n
6 *=
2L Y.
[ 1i=
Our estimate for DCp is obtained by correcting the estimates
A and for bias and noting that 9711=n onl
y
j
exists for values of N>2.
CHAPTER IV
DENSITY ESTIMATION FOR CLUSTERED POPULATIONS
4.1 Introduction
The estimation procedures developed in Chapters II and
III are based on the assumption that the sightings of ani
mals are independent events. These methods would be appli
cable to animal populations that are generally made up of
solitary individuals, such as ruffed grouse, snowshoe hare
and gila monster. However, there are other types of animals
which aggregate into coveys, schools and other tight groups.
Animals behaving in this way will be said to belong to clus
tered populations. Some examples of clustered populations
are bobwhite quail, gray partridge and porpoise. In these
cases the assumption of independent sightings is certainly
not valid, and a different procedure would have to be used.
The line transect method could be easily generalized to
provide estimates for clustered populations. As noted by
Anderson et al. (1976, p. 12), if we amend the assumptions
in Section 2.2.1 so that they refer to clusters of animals
rather than individual animals, then the results of Chap
ters II and III are directly applicable to the estimation
of the cluster density, Dc. The estimate for Dc will be
71
based on the right angle distances to the clusters from the
random line transect. In the case where the number of ani
mals in every sighted cluster can be determined without error,
an estimate for the population density D is given by
D = D s
where Dc is the estimate for Dc and s is the average size
of the observed clusters.
Some criticisms of the approach outlined in the preced
ing paragraph are possible. First of all, it may not be
possible to determine the distance to a cluster as easily
(or as accurately) as the distance to an animal. How will
this distance be defined? Secondly, the simple modification
of the assumptions in Section 2.2.1, obtained by replacing
the word "animal" by the word "cluster" would imply that the
probability of sighting a cluster depends only on its right
angle distance from the line. This may not be a reasonable
assumption since the probability of sighting a larger cluster
is likely to be greater than the probability of sighting
a smaller cluster. Finally, the sighting of a cluster may
not necessarily mean that all of the animals comprising the
cluster are seen and counted by the observer. In this case,
a more reasonable assumption would be to let the probability
of sighting an animal belonging to a cluster depend on the
distance to the cluster as well as the true cluster size.
In this chapter we shall propose a density estimate for
a clustered population by assuming, among other things, that
it is possible to determine the distance to the center of the
cluster from the line transect. An estimation procedure will
then be developed using a model in which the observer's count
of the number of animals in a cluster is regarded as a random
variable with a probability distribution depending upon the
right angle distance and the size of the cluster.
4.2 Assumptions
The density estimate that we will develop is based on
the inverse sampling approach outlined in Section 2.1, with
one minor modification. In clustered populations the plan
is to continue sampling along a randomly placed cransect
until a prespecified number, Nc, of clusters (rather than
animals) are seen. As each cluster is sighted, the follow
ing information is recorded:
1. the right angle distance, y, from the transect
to the center of the cluster
2. the observed number of animals, s, in the cluster.
(this may be less than the true size of the cluster)
3. the actual distance, Z, travelled by the observer
to sight N clusters.
The sampling procedure described above may be used to
construct an estimate of the population density under the
following set of assumptions. These assumptions closely
parallel those of Section 2.2.1 with the exception chat
they are now phrased in terms of clusters rather than indi
vidual animals.
Bl. The clusters are randomly distributed with rate
(density) D over the area of interest, A.
B2. The clusters are independently distributed over A,
i.e., given two disjoint regions of area, 6A1 and
6A2'
P(n1 clusters are in .A1 and n2 clusters are in 6A2)
= P(nI clusters are in 6A1)P(n2 clusters are in 6A2).
B3. Clusters are fixed, i.e., there is no confusion
over clusters moving during sampling and none are
counted twice.
B4. There exists a probability mass function p()
defined on the set of positive integers, such that
p(r) is the probability that r is the true size
of a cluster located at a right angle distance, y,
from the transect. Note that p(r) is independent
of y. In probability notation, if R and Y denote
the random variables representing the true cluster
size and the right angle distance to the cluster,
respectively, then
P(R=rjY=y) = p(r), r = 1,2,... (4.1)
B5. The probability of observing a cluster depends
only on the size of the cluster and the distance
from the transect to the cluster.
B6. There exists a nonnegative function h() defined
on 10,w) such that
0 5 h(.) < 1,
h(0) = 0,
and the probability of observing s animals belong
ing to a cluster of size r 2 s located at a right
angle distance y from the transect is
(r) [h(y) ls lh(y) rs
That is, if Y and S denote the random variables
represent ng tLhe right angle di:sLtance to a cluster
and the observed number of animals in a cluster,
respectively, then
P(S=slR=r,Y=y) = (r)[h(y)Is[h(y) rs. (4.2)
\s
Closer examination of assumption B6 shows that we are
now allowing the probability of observing a cluster to depend
on both the right angle distance, y, and the true cluster
size, r. To see this, first let C be the event that a cluster
is observed. Then the probability of observing a cluster of
size r located at distance y from the transect is
r
P(CIR=r,Y=y) = E P(S=slR=r,Y=y)
s=l
= 1 P(S=0IR=r,Y=y)
= 1 [lh(y)]r, (4.3)
which clearly depends on both y and r.
The assumption B6 also satisfies the reasonable require
ment that for a fixed right angle distance y >0 and r1 < r2,
P(CIR=rl,Y=y) P (C R=r2,Y=y).
This follows immediately from equation (4.3). Note that
P(CIR=rl,Y=y) = 1 [lh(y)] l
and
P(CIR=r2,Y=y) = 1 [lh(y)]2
Now, since 0 h(y) 1, it is clear that
P(CIR=rl,Y=y) _P(CIR=r2,Y=y).
One final note with regard to assumption B6 is in order.
In the case where every cluster has size 1, i.e.,
P(R= 1) =1,
the probability of sighting a cluster located at a right
angle distance y is simply h(y). This is quickly seen by
setting r= 1 in (4.3). Thus, under these circumstances,
h(y) has the same interpretation as g(y) defined in Sec
tion 2.2.1, that is, h(y) is the conditional probability of
sighting an animal at distance y given there is an animal
at v.
4.3 General Form of the Likelihood Function
We will use the maximum likelihood procedure to obtain
an estimate for D, the animal population density. To obtain
the likelihood function, we first need an expression for the
probability density function
fS,Y,L s'y' )
where S= (S1,S2, .... SN ) is the vector of random variables
c
representing the actual number of animals seen in the clusters,
Y= (Yi ,\Y ...2' Y ) is the vector of random variables repre
c
senting the right angle distances from the clusters to the
transect and L is the random variable representing the total
length travelled on the transect to see H clusters. Upon
writ ing
fS ( = fS I 'Y, L(s)I'y) f [, L 1)fL (9(), (4.4)
S. 'i L S YSI^,LilL L
it is seen that specifying the joint probability density
function for S,Y and L is equivalent to specifying the three
functions on the right hand side of (4.4).
The density functions fyIL(ylk) and fL(k) may be derived
in a manner analogous to that used in Section 2.2. Let g (y)
denote the probability of sighting a cluster located at a
right angle distance y from the transect, that is
gc(y) =P(observe a cluster Y=y).
Since sighting a cluster located at a distance y is equiv
alent to observing at least one animal belonging to the
cluster, we can write
gc(y) = E P(observe s animals Y=y)
s=l
= E P(S=s Y=y).
s=l
Now, for s l1,
P(S=slY=y) = E P(S=slR=r,Y=y)P(R=rlY=y).
r=s
By assumption B4, it follows that Y and R are independent
random variables. Thus, using (4.1) and (4.2) we get
P(S=sIY=y) = Z s (h(y)lslh(y)rs p(r).
r=s
Therefore,
g (y) = s ( [h(y)s [lh(y)rs p(r). (4.5)
s=1 r=s
Now, according to assumption B6
h(0) = 1,
so that
g (0) = p(s) =1.
s=1
Therefore, the function gc(y) plays a role similar to the
role of g(y) in Section 2.2. Consequently, by regarding a
"cluster" as an "animal," the results of Section 2.2 can be
applied to clustered populations in a straightforward manner.
Let Nc () denote the random variable representing the
number of clusters seen when travelling a distance on the
transect. Then, by Theorem 1, Nc () is a Poisson process
with parameter 0*L, where 0* is the expected number of
clusters seen when travelling along a transect of length Z.
Also, from Theorem 1 we see that the respective analogs to
equations (2.2) and (2.1) are
0
D (4.6)
c 2c
where Dc is the density of clusters and
Cr
c' = gc y)dy. (4.7)
Furthermore, Theorem 2 gives us the results that L and Y are
mutually independent random variables, L is distributed as
a Gamma random variable with parameters N and 0 and the
conditional density of Y given L= R is
N
IL( fY) IT c gc(Yi). (4.8)
(c*) c i=l
Now, assumption B5 implies that the number of animals
actually observed in a cluster depends only on the right
angle distances to the animals, Y, and the size of the
cluster, R. Thus, S is independent of L, and since Y is
also independent of L it follows that
fSIY,L (sly,) = fsly(S Y)
Nc
= n P(S =s .Y.=y.). (4.9)
i=l i
We can now write an expression for the likelihood func
tion L(e*,p(.),h(.);s,y,t). Using (4.4), (4.8) and (4.9)
and recalling that L has a Gamma distribution with param
eters N and 0 we obtain
c
N
L(6*,p(.),h(.);s,y,t) = H P(S=sjY=y)
N N Nl 
H gc(Yi)(O*) c Z c e
x i= (4.10)
(c*) c F(N )
4.4 Estimation of D when p(*) and h(.)
Have Specific Forms
For a clustered population with a cluster density, D ,
the animal population density may be defined as
D=Dc ,
where v =E(R) is the expected cluster size. Upon using the
expression for D given in (4.6), we get
0*v
D= 0 (4.11)
2c
so that maximum likelihood estimation of D can be carried out
by using (4.11) in the likelihood function presented in (4.10).
Since the random variables S and Y are independent of L,
it is easily seen from (4.10) that the maximum likelihood
estimate of O' corrected for bias is
1
S (4.12)
However, finding estimates of v and c can be quite difficult
depending upon the nature of the functions p(.) and h().
Very likely, one has to resort to some iterative technique
such as the NewtonRaphson method (see Korn and Korn, 1968,
eqn. (20.231)) to solve the likelihood equation.
It is apparent that there exist a wide variety of func
tions which satisfy the requirements of p() and h().
The appropriate choice in a particular problem depends on
the nature of the population under investigation. In this
work we will consider the functions
r C
p(r) = c e a > 0, r = 1,2,... (4.13)
rl(1e )
and
h(y) = e > 0, v > 0. (4.14)
It is easily seen that p(.) given by (4.13) represents a
truncated Poisson distribution. The expected cluster size
v is therefore given by
SC (4.15)
le
The limiting case a = 0 corresponds to a population in which
the cluster size is 1 with probability 1. Thus, c =0 corre
sponds to the model in Section 2.2.
The choice for h() is based on the fact that when a=0,
h() may be interpreted as the function g(*) defined in
Chapter II. Because
g(y) = e y
seems to be a popular choice for g(*), we feel that
h(y) = e y
is a reasonable choice for h().
The likelihood function may now be regarded as a function
of 0*, a and X*, and maximum likelihood estimation of
D 
D 0 *
2c*
may be accomplished by expressing v and c as functions of
a and A We have already seen the form of v in equation
(4.15). To derive an expression for c we proceed as follows.
Recall from (4.5) that
g (y) = E P(S=sIY=y),
s=l
where
P(S=sY=y) = s) [h(y)]s[1h(y)]rs p(r).
r=s
Now using (4.13) and (4.14) in the above equation, we get
( s
r\ 4 yst A 'k f r a
s e e
P(S=sJY=y)= E s
r=s rl(1e)
(ae ~ys *y rs
S(ae y)s e [ra (1e y)]
e ) (rs)1
* A y
(ee y)s e"
s(1ees (4.16)
sl(le )
Then, substituting for P(S=sY=y), we get
oe y A v s
g(y) = e e
(Ie ) s=l1
le
ea (4.17)
(le )
Therefore, using (4.7) and (4.17), we get
c g c(y)dy
eA
z) dy (4.18)
(Le a)
To evaluate c note that
0 \ *y
I  e )dy "lie y
(e )dy = lim { e dy) (4.19)
,3 X *+? *3o0
By letting
t = e
in the integral in the right hand side of (4.19), we can show
"" A N V 1 aC t
ae dy 1 e dt
ee dy = t
e
1 (1) aj(le ) j
V + g
A j=l j j
and upon substituting inco (4.19), we get
o e 1 0 ( 1 ) a ( L e )
(le )dy = lim L c
yo j=1 j j!
Since the sum above is absolutely convergent, we can take
the limit inside the sum and obtain
o a*e 1 J
(le )dy = ), i)
0o j=l j
Then, using (4.18), it follows that
c = a) (4.20)
S(le0)
where
a(a) = E ) (A.21)
j=l J
We can now write the likelihood function in terms of
6*, \* and a. Using equations (4.10), (4.16), (4.20) and
(4.21), we get
N N N *
c c c A Yi
E s. y.s. a e
i=l i=l i=l
a e e
L('*,X*,a;s,, ) = ;eN e
N c
(le") c H s.!
(le ) i=l
N N ae Y N N 1
( c c ae )(0i c c9.
0. ) T (1e )e
i=l
N
[a(a)] c F(Nc)
(4.22)
Using the likelihood function given in (4.22), we can
now obtain an estimate for the population density D.
Recall from (4.11) that we can write
*
0 V
D 
2c*
After substituting for c and v using equations (4.15),
(4.20) and (4.21), the expression for D becomes
D 0 ) k
2 2a(i)
Thus, an estimate for D would be
2a(a)
where 0 X and a are maximum likelihood estimates for
0 A and a, respectively. As noted earlier in this section,
S and Y are independent of L so that O0 can be estimated
using equation (4.13). However, this still leaves us the
problem of estimating A* and a. Instead of estimating A*
and a separately, we can reparamecerize the likelihood
equation in (4.22) by letting
0
and
a = a.
Then, our estimate for D becomes
D = (4.23)
The advantage of this reparameterization is that it makes
use of the fact that L is independent of both S and Y. Thus,
the estimate for D given in (4.23) is now the product of two
independent esLimates, 0 which depends on L alone and D which
depends on S and Y. As a result the variance of can now
be found easily. Using the formula (see Goodman, 1960) for
the variance of the product of two independent estimates,
we get
Var(D ) =E (6)Var(p)+E (p)Var(6) +Var(6 )Var(p)
(4.24)
Since L is distributed as a Gamma random variable with
an* (^ *
parameters Nc and 0 exact expressions for E(6 ) and Var( )
can be obtained using (2.4) and (2.11), i.e.,
E(0 ) = ,
and
*2
Var(e ) =N  (4.25)
c
Expressions for the variance and expected value of p can be
come quite complicated. An iterative scheme would be needed
to find the solutions for p and a that would maximize the
reparameterized version of the likelihood function given
in (4.22). There are computer programs available that can
provide maximum likelihood estimates for p and a along with
numerical approximations for the variance covariance matrix
of the estimates. In the next section we will demonstrate
the use of one such program with a set of hypothetical data.
4.5 A Worked Example
In this section we will present a worked example to
demonstrate the use of a computer program to find the estimate
D* and its approximate variance. Because we are not aware of
any real data that have been collected according to the sam
pling plan described in Section 4.2, we shall use an artifi
cial set of data in the example.
Suppose that sampling was continued until N = 25 clusters
were sighted, and that a transect length of = 25 miles was
needed to sight the 25 clusters. Suppose further that the
observed right angle distances and the cluster sizes were as
follows, where the first number in the pair is the right
angle distance, y, measured in yards and the second number
in the pair is the corresponding cluster size, s:
(1,1), (3,2), (7,1), (10,1), (2,3)
(5,5), (4,1), (7,2), (15,1), (22,1)
(6,1), (3,6), (2,1), (12,1), (28,3)
(9,2), (18,1), (36,7), (17,6), (5,1)
(4,1), (3,1), (8,2), (3,4), (13,1).
As noted in Seccion 4.4, an estimate for 0 is
N 1
c
c = .96,
and an estimate for the variance of ,U is
Var(0 ) = .001.
c
In order to estimate p, the reparameterized version of
the likelihood function given in (4.22) will have to be max
imized. The Fortran subroutine ZXMIN, found in IMSL (1979)
may be used for this purpose. This program uses a quasi
Newton iterative procedure to find the minimum of a function.
Thus, we first need to take the negative of the likelihood
equation before we can use this subroutine to our advantage.
On output, this subroutine not only provides the values
at which the function is minimized, but also provides numer
ical estimates for the second partial derivatives of the
function evaluated at the minimization point. Thus, when
used with the negative of the likelihood function this pro
gram will provide the maximum likelihood estimates, p and i,
as well as the matrix of negative second partial derivatives
of the likelihood, L(*), evaluated at p and a. We will denote
this matrix by
2 2
/ 2IL(*) 2 ZnL()
2
1
a2nL() aInL ( ) /a=a
\ a 9p p2 /
For our data, the use of the subroutine ZXMIN with
initial values ai = 2.24 and pI = .16 yielded
a = 2.844,
p = .0907,
and
88
S 7.687 161.229
161.229 5098.985
The initial value used for a was the mean of the observed
cluster sizes, i.e.,
25
r S.
i=l 
L 25 = s.
25
Since our model does not assume all animals belonging to a
cluster are seen, s would underestimate the expected cluster
size, i.e.,
s < E(R) .
1e
Thus, s seems to be a good starting value for a.
In choosing an initial value for P, first recall that
0 = a(O ,
where a(a) is given in equation (4.21). Since our initial
value for a is s, all we need is a starting value for A.
If every animal in the cluster was seen with probability 1,
the density of clusters would be estimated by the method
described in Chapter II. In this case, the maximum likelihood
estimate for .\ would be 1/7 where
25
y' =
Thus, as the initial value for p we used
S
1 
ya(s)
The estimate for the density can now be calculated.
Using (4.23) and substituting the values we obtained for
6 and p, we get
A*
D = 76.7 animals/square mile.
Now if we can obtain a large sample approximation for
the variance of p, then we can use (4.24) as an approxima
tion for the variance of D Now, under the usual regularity
conditions, V will be a large sample approximation to the
inverse of the variancecovariance matrix of a and p. Further
more, the approximate variance of D* can be obtained from
equation (4.24) after substituting the element in the matrix
corresponding to the approximate variance of p along with
the other appropriate quantities. Straightforward calcula
tions show that
SVar(D*) 26.2 animals/square mile.
The use of this Fortran subroutine required a minimal
amount of programming to enter the appropriate likelihood
function. It was run using the computer facilities of the
Northeast Regional Data Center located in Gainesvillc, Florida.
Less than two seconds of CPU time was needed for the estimates
to converge to values that agreed to four significant digits
on two successive interations.
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BIOGRAPHICAL SKETCH
John Anthony Ondrasik was born on August 17, 1951, in
New Brunswick, New Jersey. Shortly thereafter his parents
moved to Palmerton, Pennsylvania, where he grew up and
attended high school. After graduation in June, 1969, he
entered Bucknell University in Lewisburg, Pennsylvania, and
received the degree of Bachelor of Science with a major in
mathematics in June, 1973.
It was during his studies at Bucknell that he became
interested in statistics through the influence of the late
Professor Paul Benson. In September, 1973, he matriculated
in the graduate school at the University of Florida and
received the degree Master of Statistics in 1975.
. lile pursuing his graduate studies, he worked for the
Department of Statistics as an assistant in their biosta
tistics consulting unit. In November, 1978, he accepted the
position of biostatistician with Boehringer Ingelheim, Ltd.
John Ondrasik is married to the former Anntoinette M.
Lucia. Currently they reside in Danbury, Connecticut.
