Local central limit theorems, the high-order correlations of rejective sampling and logistic likelihood asymptotics

Abstract

Let I₁,…,I_n be independent but not necessarily identically distributed Bernoulli random variables, and let X_n=∑_j=1ⁿI_j. For ν in a bounded region, a local central limit theorem expansion of $\mathbb {P}(X_{n}=\mathbb {E}X_{n}+\nu)$ is developed to any given degree. By conditioning, this expansion provides information on the high-order correlation structure of dependent, weighted sampling schemes of a population E (a special case of which is simple random sampling), where a set d⊂E is sampled with probability proportional to ∏_A∈dx_A, where x_A are positive weights associated with individuals A∈E. These results are used to determine the asymptotic information, and demonstrate the consistency and asymptotic normality of the conditional and unconditional logistic likelihood estimator for unmatched case-control study designs in which sets of controls of the same size are sampled with equal probability.