Sequential Maximum Likelihood Estimation of the Size of a Population

Abstract

Consider the following model, the practical applications of which will be discussed elsewhere. An urn contains an unknown number, $N$, of white balls, and no others. An estimate of $N$ is desired, based on the following sampling procedure. Balls are drawn at random, one at a time, from the urn. A white ball is colored black before it is returned, a black ball is returned unchanged. The ball is always returned before the next ball is drawn. We are interested in two problems: (i) what stopping rule $t$ to use to terminate sampling, and (ii) how to estimate $N$ after we stop. The present problem (also in a more general setup) has been considered by several authors, notably L. A. Goodman [5], Chapman [1], Darroch [3] and Darling and Robbins [2]. We shall refer to their results in the sequel. Let $w_i, b_i$ denote the (random) number of white balls, black balls, respectively, observed in the first $i$ draws $(w_i + b_i = i)$. We shall consider mainly the following stopping rules. RULE A. Let $A > 0$ be a fixed integer. $t_A = A$. RULE B. Let $B > 0$ be a fixed integer. $t_B = \inf \{i\mid b_i = B\}$. RULE C. Let $C > 0$ be fixed. $t_c = \inf \{i|b_i \geqq Cw_i\} = \inf \{i|i \geqq (C + 1)w_i\}$. RULE D. Let $- \infty 0$ for all $N \geqq A$. Rule B is discussed in Section 4 and it is shown that $2B\hat{N}/N$ has an asymptotic chi square distribution with $2B$ degrees of freedom, (to be denoted $\chi^2_{2B})$, as $N \rightarrow \infty$. In Section 5 Rule C is considered, and bounds on the distribution of $(\hat{N} - N)N^{-\frac{1}{2}}$ in terms of the normal distribution are given. Rules D and E are considered in Section 6. Let $\lbrack x\rbrack^\ast$ be the largest integer not exceeding $x$. For Rule D it is shown that $\hat{N} - N + \lbrack\lambda\rbrack^\ast$ has an asymptotic Poisson distribution with parameter $\lambda = \exp (-D - 1)$ and for Rule E $P_N(\hat{N} = N) \rightarrow 1$. The exact and asymptotic distributions of the corresponding $t$'s is also considered, and is closely related to the distribution of $\hat{N}$.