Abstract
Let I1,…,In be independent but not necessarily identically distributed Bernoulli random variables, and let Xn=∑j=1nIj. 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∈dxA, where xA 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.
Citation
Richard Arratia. Larry Goldstein. Bryan Langholz. "Local central limit theorems, the high-order correlations of rejective sampling and logistic likelihood asymptotics." Ann. Statist. 33 (2) 871 - 914, April 2005. https://doi.org/10.1214/009053604000000706
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