Large sample theory of maximum likelihood estimates in semiparametric biased sampling models

• 成果类型: Article
• 署名作者: Gilbert, PB
• 署名单位: Harvard University; Harvard T.H. Chan School of Public Health
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1016120368
• 发表日期: 2000
• 页码: 151-194
• 关键词: choice-based samples; empirical distributions

摘要

Vardi [Ann. Statist. 13 178-203 (1985)] introduced an s-sample biased sampling model with known selection weight functions, gave a condition under which the common underlying probability distribution G is uniquely estimable and developed simple procedure for computing the nonparametric maximum likelihood estimator (NPMLE) G(n) of G. Gill, Vardi and Wellner thoroughly described the large sample properties of Vardi's NPMLE, giving results on uniform consistency, convergence of root n (G(n) - G) to a Gaussian process and asymptotic efficiency of G(n). Gilbert, Lele and Vardi considered the class of semiparametric s-sample biased sampling models formed by allowing the weight functions to depend on an unknown finite-dimensional parameter theta. They extended Vardi's estimation approach by developing a simple two-step estimation procedure in which <(theta)over cap>(n) is obtained by maximizing a profile partial likelihood and G(n) = G(n)(<(theta)over cap>(n)) is obtained by evaluating Vardi's NPMLE at <(theta)over cap>(n). Here we examine the large sample behavior of the resulting joint MLE (<(theta)over cap>(n), G(n)), characterizing conditions on the selection weight functions and data in order that (<(theta)over cap>(n), G(n)) is uniformly consistent, asymptotically Gaussian and efficient. Examples illustrated here include clinical trials (especially HIV vaccine efficacy trials), choice-based sampling in econometrics and case-control studies in biostatistics.