Asymptotic Theory of Rejective Sampling with Varying Probabilities from a Finite Population

Abstract

In [3] the author established necessary and sufficient conditions for asymptotic normality of estimates based on simple random sampling without replacement from a finite population, and thus solved a comparatively old problem initiated by W. G. Madow [8]. The solution was obtained by approximating simple random sampling by so called Poisson sampling, which may be decomposed into independent subexperiments, each associated with a single unit in the population. In the present paper the same method is used for deriving asymptotic normality conditions for a special kind of sampling with varying probabilities called here rejective sampling. Rejective sampling may be realized by $n$ independent draws of one unit with fixed probabilities, generally varying from unit to unit, given the condition that samples in which all units are not distinct are rejected. If the drawing probabilities are constant, rejective sampling coincides, with simple random sampling without replacement, and so the present paper is a generalization of [3]. Basic facts about rejective sampling are exposed in Section 2. To obtain more refined results, Poisson sampling is introduced and analyzed (Section 3) and then related to rejective sampling (Section 4). Next three sections deal with probabilities of inclusion, variance formulas and asymptotic normality of estimators for rejective sampling. In Section 8 asymptotic formulas are tested numerically and applications to sample surveys are indicated. The paper is concluded by short-cuts in practical performance of rejective sampling. The readers interested in applications only may concentrate upon Sections 1, 8 and 9. Those interested in the theory of mean values and variances only, may omit Lemma 4.3 and Section 7.