Unbiased Sequential Estimation for Binomial Populations

Abstract

The subject of minimum variance unbiased estimation has received a great deal of attention in the statistical literature, e.g., in the papers of Bahadur [2], Barankin [3], and Stein [14]. The emphasis in these papers has typically been placed on the existence and construction of minimum variance unbiased estimators when the sampling plan to be used was given in advance. In this paper, criteria are developed for the selection of an appropriate sampling plan for the family of binomial distributions. Thus, independent observations are to be taken on the random variable $U$ so distributed that \begin{equation*}\tag{1.1} \operatorname{Pr} \{U = 1 | p\} = p,\quad \operatorname{Pr} \{U = 0 | p\} = 1 - p = q,\end{equation*} where $p$ lies in the open interval $0 < p < 1$, and the value of a given function, $g(p)$, is to be estimated. The problem considered here is that of determining a sampling plan and an unbiased estimator of $g(p)$ that are optimal, in some sense, at a specified value, $p_0$, of $p$. Optimality will depend, not only on the variance of the estimator, but also on the average sample size of the plan. A sampling plan, $S$, and estimator, $f$, will be considered optimal at $p_0$ if, among all procedures with average sample size at $p_0$ no larger than that of $S$, there does not exist an unbiased estimator with smaller variance at $p_0$ than that of $f$. The basic tool to be used is the information inequality (see Lemma 2.7 and the discussion following it) which provides a lower bound for the variance of an estimator in terms of its expected value and the average sample size of the sampling plan. If, at $p_0$, this lower bound is attained for a particular estimator and sampling plan, it may be immediately concluded that they are optimal at $p_0$. Such an estimator is said to be efficient at $p_0$. In Section 2, various definitions, assumptions, and fundamental facts to be used throughout the paper are collected. In Section 3 it is shown that the single sample plans and the inverse binomial sampling plans are the only ones that admit an estimator that is efficient at all values of $p$. In Section 4 some techniques are given that are often useful in the analysis of inverse binomial sampling plans. In Section 5 relationships between the average sample size of a sampling plan and the functions of $p$ that are estimable optimally are explored. In Sections 6 and 7 it is shown that there can exist two distinct sampling plans with the same average sample size for all $p$ and some comparisons are made of such plans. In Section 8 a new characterization of completeness is given for bounded sampling plans and it is shown that the dimension of the linear space of unbiased estimators of 0 can be determined simply by counting the number of boundary points. It is further shown that for a wide class of plans, the estimators that are efficient at a given value of $p$ do not have uniformly minimum variance, although non-trivial uniformly minimum variance estimators do exist. After this paper had been accepted for publication I learned that R. B. Dawson had obtained expression (4.13) and Theorems 8.2 and 8.4, as well as various other interesting results related to the material of Section 8, in his Ph.D. thesis, "Unbiased tests, unbiased estimators, and randomized similar regions," Harvard University, May, 1953.