Local central limit theorems, the high-order correlations of rejective sampling and logistic likelihood asymptotics
Richard Arratia, Larry Goldstein, and Bryan Langholz
Source: Ann. Statist. Volume 33, Number 2 (2005), 871-914.
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 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.
Primary Subjects: 62N02, 62D05, 60F05, 62F12
Keywords: Case-control studies; epidemiology; frequency matching
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1117114339
Digital Object Identifier: doi:10.1214/009053604000000706
Mathematical Reviews number (MathSciNet):
MR2163162
Zentralblatt MATH identifier:
1068.62106
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