Limiting Behavior of High Order Correlations for Simple Random Sampling

TL;DR

This paper investigates the asymptotic behavior of high order correlations in simple random sampling, proving a conjecture about their limiting distribution as the population size grows large.

Contribution

It establishes the limiting behavior of high order correlations in simple random sampling, confirming a previously conjectured asymptotic form.

Findings

High order correlations depend only on the subset size k.

Correlations scale with N^{k/2} or N^{(k+1)/2} depending on parity of k.

Limit distributions involve moments of a standard normal variable.

Abstract

For N=1,2,..., let S_N be a simple random sample of size n=n_N from a population A_N of size N, where 0<=n<=N. Then with f_N=n/N, the sampling fraction, and 1_A the inclusion indicator that A is in S_N, for any H a subset of A_N of size k>= 0, the high order correlations Corr(k) = E (\prod_{A \in H} (1_A-f_N)) depend only on k, and if the sampling fraction f_N -> f as N -> infinity, then N^{k/2}Corr(k) -> [f(f-1)]^{k/2}EZ^k, k even and N^{(k+1)/2}Corr(k) -> [f(f-1)]^{(k-1)/2}(2f-1)(1/3)(k-1)EZ^{k+1}, k odd where Z is a standard normal random variable. This proves a conjecture given in [2].