AbstractThe size of a closed population is to be estimated using data from a multiple recapture study in either continuous or discrete time. Here the use of maximum likelihood raises computational problems. However, a family of martingale estimating functions related to the score function is shown to produce convenient simple estimators with good asymptotic efficiency relative to the maximum likelihood estimator

Becker, N.G.

Heyde, C.C.

Elsevier - Publisher Connector 

Stochastic Processes and their Applications 36 (1990) 77-83 
North-Holland 
77 
ESTIMATING POPULATION SIZE FROM MULTIPLE 
RECAPTURE EXPERIMENTS 
N.G. BECKER 
Department of Statistics, La Trobe University, Bundoora, Vie. 3083, Australia 
C.C. HEYDE 
Statistics Research Section, IAS, Australian National University, GPO Box 4, Canberra, 
ACT 2601, Australia 
Received 6 February 1989 
Revised 4 December 1989 
The size of a closed population is to be estimated using data from a multiple recapture study in 
either continuous or discrete time. Here the use of maximum likelihood raises computational 
problems. However, a family of martingale estimating functions related to the score function is 
shown to produce convenient simple estimators with good asymptotic efficiency relative to the 
maximum likelihood estimator. 
population size * capture-recapture * estimating function * optimal estimation * quasi-Welihood 
* martingale * stochastic integral. 
1. Introduction 
Consider a population in which there are Y animals, where Y is unknown. A multiple 
recapture experiment is conducted in which animals are marked at the time of their 
first capture. At each subsequent capture the mark is correctly recorded. Apart from 
time allowed for marking or noting of marks, each captured animal is released 
immediately. The aim is to make inference about the parameter v from the set of 
observed captures of marked and unmarked animals. 
Results are derived using a continuous-time formulation. Analogous results hold 
for multiple recapture experiments in discrete time and these are summarized briefly 
in Section 6. 
Label the animals by 1,2, . . . , v and let Ni( t) denote the number of times animal 
i has been caught in [0, t]. Write N, for (N,(t), . . . , ND(t))‘. Each {Ni(t); t> 0) is 
a continuous-time counting process with right-continuous sample paths and common 
intensity function A defined by 
0304-4149/90/$3.50 @ 1990-Elsevier Science Publishers B.V. (North-Holland) 
78 N.G. Becker, CC. Heyde / Estimating population size 
where .!Fr is the u-field generated by {NX ; 0 s x s t}. The dependence of A on t can 
reflect the dependence of the animal’s behaviour on time as well as variations in 
the intensity of trapping. 
2. Maximum likelihood estimation 
Consider observations from a multiple capture experiment over [0, T]. The total 
number of captures by time t is given by N, =CyZI Ni(t). Write M, for the number 
of animals that are marked by time t. Then M, is the number of captures of unmarked 
animals, while K, = N, - M, gives the number of captures of marked animals by 
time t. 
A direct approach to maximum likelihood estimation of I/ based on observation 
of {M,, N,, 0~ t G T} is complicated by the fact that the intensity function A, is 
generally unknown. We find that the log-likelihood function is 
L= 
I 
7 
log(b) Wx - v I,:,dx+[; log( v - My-) d M, 
cl 
I 7 i 
= 
I 
log(h:) dN, - 
I 
Azdx-N,logv+ 
I 
log( v - M,_) dM,, 
0 0 0 
where A z = VA, is an arbitrary positive value if A, is arbitrary positive. The likelihood 
function is maximized with respect to v by the solution of 
dL --= v 
LiV 
_lN,- 
’ 
(1) 
irrespective of the form of A*. We refer to this as the maximum likelihood equation 
for v. See Samuel (1969), and references quoted by her, for more detailed consider- 
ation of related maximum likelihood estimation. 
Estimation of v via equation (1) requires the use of a computer because of the 
term v-M,_ in the denominator. We now derive estimators for v which have 
explicit expressions, are easy to compute, and have high asymptotic efficiency relative 
to the benchmark provided by (1). 
3. A class of estimators 
To facilitate avoidance of the computationally troublesome (v-M,-) in the 
denominator of (1) it is sensible to work with the martingale H, defined by 
dH, = M,- dM, - (v-M,-) dK, , (2) 
which is suggested by the right hand alternative representation for the martingale 
N.G. Becker, C.C. Heyde / Estimating population size 19 
and to seek effective estimating functions from within the class of martingale 
estimating functions 
U 
i 
fX dHX,fX predictable . (4) 
0 1 
The general question of finding an optimal estimating function within such classes 
is discussed in detail in Godambe and Heyde (1987) and Heyde (1987). The best 
choice, termed the quasi-score estimating function, is 
I 
7 (dHJ(d(H),)-’ dH,, (5) 
0 
where 
dt7, = E(d(dHJdv) .%_) = -E(dK,I FX-) = -M,- dA, 
and (H), is the quadratic characteristic of the martingale H,, so that 
d(H), = E((dHX)*lSX_) = E(M:_ dM,+(v-M,_)* dK,I SX-) 
= vM,_( v - IV&-) d/l,, 
and (5) corresponds to (3). Thus the quasi-likelihood estimator is the maximum 
likelihood estimator (MLE), v^ say. 
Now the martingale information in (3) (Heyde, 1987), whose reciprocal is essen- 
tially an asymptotic variance, is 
I -1 ML=” 
I 
T Mx d& 
0 v-M,’ 
and the central limit results of Aalen (1977) and Rebolledo (1980) can be used to 
show that 
tends in distribution to the standard normal law as v + CO. The quantity I,, behaves 
asymptotically as the observed Fisher information and martingale information is 
used for comparative purposes, being defined for all martingale estimating functions. 
Maximum likelihood provides a benchmark but other estimating functions from 
the family (4) of the type 
i 
7 
g, dH,, (7) 
0 
where g, is of the form g(N,_), produce simple and easily computable estimators 
I 
7 
/I 
7 
5, = gX,- dN, g, dK, (8) 
0 0 
whose performance relative to the benchmark is well worth considering. We shall 
examine a number of choices for g, and, in particular, determine whether g, = 1 is 
a good choice. 
80 N.G. Becker, C.C. Heyde / Estimating population size 
4. Asymptotic relative efficiency 
The martingale information in the estimating function (7) is 
and, subject to suitable restrictions on g, the above cited central limit results can 
be used to show that 
tends in distribution to a standard normal law as v + ~0. To investigate the relative 
behaviour of the random quantities (6) and (9) as v increases, information on the 
distributions of M, and N,, x > 0, is needed and one can show that the probability 
generating function of (M,, Nx) is 
E( y”‘zNx )=e-““~(l-y+ye’“~)“. 
Thus, M, is distributed as Binomial( v, 1 -e-“v) and N, as Poisson( VA,). In par- 
ticular, V-’ M, a.s: 1 -e-*x and V-’ N, 4 A, as v + 00. 
Then, if the function g is well behaved, and, in particular, there exists a sequence 
of constants {c,} such that 
P 
WL. - &i (xl, x E LO, 71, 
as v + CO, where gd (x) is a deterministic function, then it follows that 
(10) 
and 
P 7 
VI, - 
(I 
2 7 
g,(x)(l -e-“.x) dA, 
0 )/I 
g:(x) e-“=( 1 -e-“r) dA,. 
0 
The asymptotic relative efficiency of Gg to the benchmark of the MLE is the ratio 
A=( 6%) = (e-A 
(j,‘gd(x)(l -ep”r) dAJ2 
r-l-A,)j,‘g:(x)e-“x(1-e-“r)dA,’ 
(11) 
That ARE( 6,) is less than or equal to unity follows from the Schwarz inequality 
and equality holds iff gd (x) = exp A,. 
5. Convenient and efficient estimators 
There is no obvious choice for g which satisfies the requirement (10) for gd(x) = 
N.G. Becker, C.C. Heyde / Estimating population size 81 
expA, and does not involve V. Using (S), we consider the convenient estimators 
~,=~~M_dN,lI:dK,. 
;,=5:M:~dN~/~~M,_dK,, 
and 
l 
7 
/I 
I 
& = M,_N,_ dN, N,_ dK, 
0 0 
for which the form of gd(x) in (10) is 1, 1 -e-“v and A, respectively. By direct 
integration, 
ARE( ;,) = 
2(A,+e-“T-1)2 
(e”T - 1 -A,)(1 -e-AT)2’ 
ARE( C2) = (2A + 1 - (2 - e-nT)2)2 
(e”T - 1 -A,)(1 -e-Ar)4’ 
and 
ARE( &) = 
(A:+2A, e-*7-2+2 ep’iT)2 
(e”T - 1 -A,){3+2 e-“r(A,+ l)(A, e-“~-2A,-2)+(2-e~“~)*}’ 
The graphs of ARE( 6,), ARE( c2) and ARE( ;,) are plotted against 1 - e-“T in Fig. 1. 
This choice of scale in the horizontal axis is made because 1 -e-“7 represents the 
expected fraction of animals marked by time 7 and therefore has direct experimental 
relevance. 
From Fig. 1 it is clear that all three estimators are very efficient over the range 
0 < 1 - e-“r < 0.5, which seems to contain the range of most practical importance. 
The efficiency of G, is highest over this range, which suggests that it is better to have 
0.70 
0.65 \ 
0.60 
0.55 
\ ARE( 6,) 
0.501 I I I I I I I I I I I 1 1 I i 
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I .o 
1 - e-% 
Fig. 1. Graph of the asymptotic relative efficiency of estimators t,, i, and i, against the expected 
proportion of marked animals at the end of the experiment. 
82 N.G. Becker, C.C. Heyde / Estimating population size 
a function g in (8) which is not zero initially. The efficiency of c1 drops significantly 
once the expected fraction of marked animals exceeds 0.5, as might be expected 
since g, = 1 is not an increasing function. These observations encourage us to 
consider the estimator 
Corresponding to this estimator we have 
A 
gd(x) = exp x ( > l_e-“X ’ 
which is an increasing function starting at the value e and is close to the optimal 
enx for large A,. The efficiency of this estimator was computed by numerical 
integration. ARE( cd) was 30.995 over the range 0 < 1 -e-“7 < 0.5 and never fell 
below 0.9774. 
6. Multiple recaptures in discrete time 
In a discrete-time setting the intensity function A is replaced by the set of hazard 
probabilities P,, t E { 1,2, . . .}, given by 
p,=Pr[Ni(t)= N,(t-l)+ll%_,]. 
In this case we work with the martingale 
H, = f: M,_, dM, - i (v-M,_,) dK,, 
x=1 X=1 
where dM, and dK, denote the numbers of captures at time x which are marked 
and unmarked respectively. This is the analogue of (2). By proceeding as in Section 
3 we arrive at estimators of the form 
Gg = j, gx-,Mx-, dNx I i gx-1 Kc, x=1 (12) 
where g, is g(N,, Mx) or g(N,). We obtain the discrete time analogues of c,, &, 
9, and cd, respectively, by substituting 1, M,, N, and exp(NXl{MX,ol/MX) for g, in 
equation (12). 
Let 
which is the analogue of e-*x. Corresponding to (11) we find 
r (I- a,)I& 
ARE(cg) = c ( > <c:=, &(X - 1x1- Qx-dPJ2 .X=1 Q x c:=, da - 1x1- CLl)OXPX’ 
N. G. Becker, C. C. Heyde / Estimating population size 83 
The efficiency is unity when gd (x - 1) = 0;‘. It follows that the discussion concerned 
with the efficiency of estimators c,, &, G and c4 remains relevant for the discrete-time 3 
analogues of these estimators. 
7. Discussion 
The estimators C,, v^,, G3 and C4 are seen to be very efficient over the range of 
practical relevance. However, there are many estimators which are inefficient. One 
can obtain inefficient estimators by substituting into (8) a function g which tends 
to decrease. 
The argument leading to estimators given by (8) remains valid even when h is 
random. For example, random fluctuations in environmental factors might affect 
the trapping rate. 
If a single trap is used which captures just one animal at a time then the estimators 
given by (8) remain valid even when immediate release of the animals is not possible. 
Occupancy of the trap simply has the effect of suspending the entire capture process 
temporarily. 
References 
0.0. Aalen, Weak convergence of stochastic integrals related to counting processes, Z. Wahrsch. Verw. 
Gebiete 38 (1977) 261-277. Correction: ibid. 48 (1979) 347. 
P.K. Andersen and 0. Borgan, Counting process models for life history data: a review, Stand. J. Statist. 
12 (1985) 97-158. 
N.G. Becker, Estimating population size from capture-recapture experiments in continuous time, Austral. 
J. Statist. 26 (1984) 1-7. 
V.P. Godambe and C.C. Heyde, Quasi-likelihood and optimal estimation, Int. Statist. Rev. 55 (1987) 
23 l-244. 
C.C. Heyde, On combining quasi-likelihood estimating functions, Stochastic Process. Appl. 25 (1987) 
281-287. 
R. Rebolledo, Central limit theorems for local martingaies, Z. Wahrsch. Verw. Gebiete 51 (1980) 269-286. 
E. Samuel, Comparison of sequential rules for estimation of the size of a population, Biometrics 25 
(1969) 517-527. 


Estimating population size from multiple recapture experiments 

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Estimating population size from multiple recapture experiments

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