Approximation of rejective sampling inclusion probabilities and application to high order correlations

Abstract

This paper is devoted to rejective sampling. We provide an expansion of joint inclusion probabilities of any order in terms of the inclusion probabilities of order one, extending previous results by Hájek (1964) and Hájek (1981) and making the remainder term more precise. Following Hájek (1981), the proof is based on Edgeworth expansions. The main result is applied to derive bounds on higher order correlations, which are needed for the consistency and asymptotic normality of several complex estimators.

Functional central limit theorems for single-stage sampling designs

“…Because these terms can be both negative and positive, they may cancel each other in such a way that (C4 * ) does hold. This is for instance the case for simple random sampling, e.g., see the discussion in Remarks (iii) and (iv) in [BO00], or for rejective sampling, see Proposition 1 in [BLRG12]. By using Lemma 2 in [BLRG12] it follows that conditions (C2 * )-(C4 * ) are implied by (C2)-(C4).…”

“…. , p N , such that N i=1 p i = n. Then, from Theorem 1 in [BLRG12] it follows that if d = N i=1 p i (1 − p i ) tends to infinity, assumption (i) holds for rejective sampling. Furthermore, if n/N → λ ∈ [0, 1] and N/d has a finite limit, then also (ii) holds for rejective sampling.…”

“…These conditions on higher order correlations are commonly used in the literature on survey sampling in order to derive asymptotic properties of estimators (e.g., see [BO00], and [CCGL10]). [BO00] proved that they hold for simple random sampling without replacement and stratified simple random sampling without replacement, whereas [BLRG12] proved that they hold also for rejective sampling. Lemma 2 from [BLRG12] allows us to reformulate the above conditions on higher order correlations into conditions on higher order inclusion probabilities.…”

“…They are implied by the usual conditions that one finds in the literature. For instance, when N/d N = O(1) (e.g., see [BLRG12]) and n/N → λ > 0, then (A2)-(A4) are immediate. Part (iii) in Corollary 5.1 is similar to Proposition 1 in [CMM15], where the Hellinger distance between P and R is used instead of (5.1).…”

“…The summands in (C4 * ) have to overcome a factor of the order N 2 , which will typically not be the case for general sampling designs. However, according to Lemma 2 in [BLRG12], the fourth order correlation can be decomposed in terms of the type…”