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

• 成果类型: Article
• 署名作者: Arratia, R; Goldstein, L; Langholz, B
• 署名单位: University of Southern California; University of Southern California
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053604000000706
• 发表日期: 2005
• 页码: 871-914
• 关键词: matched case-control; varying probabilities; relative-risk; regression; models; replacement; survival

摘要

Let I-I,..., I-n be independent but not necessarily identically distributed Bernoulli random variables, and let X-n = Sigma(n)(j=1) I-j. For v in a bounded region, a local central limit theorem expansion of P(X-n = EXn + v) 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 C E is sampled with probability proportional to Pi A is an element of d(X)A, where x(A) are positive weights associated with individuals A is an element of 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.