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April 2005 Local central limit theorems, the high-order correlations of rejective sampling and logistic likelihood asymptotics
Richard Arratia, Larry Goldstein, Bryan Langholz
Ann. Statist. 33(2): 871-914 (April 2005). DOI: 10.1214/009053604000000706

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 dE is sampled with probability proportional to ∏AdxA, where xA are positive weights associated with individuals AE. 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.

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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

Information

Published: April 2005
First available in Project Euclid: 26 May 2005

zbMATH: 1068.62106
MathSciNet: MR2163162
Digital Object Identifier: 10.1214/009053604000000706

Subjects:
Primary: 60F05 , 62D05 , 62F12 , 62N02

Keywords: Case-control studies , epidemiology , frequency matching

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 2 • April 2005
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