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February 2000 Large sample theory of maximum likelihood estimates in semiparametric biased sampling models
Peter B. Gilbert
Ann. Statist. 28(1): 151-194 (February 2000). DOI: 10.1214/aos/1016120368


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) $\mathbb{G}_n$ of $G$. Gill, Vardi and Wellner thoroughly described the large sample properties of Vardi’s NPMLE, giving results on uniform consistency, convergence of $\sqrt{n}(\mathbb{G}-G)$ to a Gaussian process and asymptotic efficiency of $\mathbb{G}_n$. 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 $\hat{\theta}_n$ is obtained by maximizing a profile partial likelihood and $\mathbb{G}_n \equiv \mathbb{G}_n(\hat{\theta}_n)$ is obtained by evaluating Vardi’s NPMLE at $\hat{\theta}_n$. Here we examine the large sample behavior of the resulting joint MLE $(\hat{\theta}_n,\mathbb{G}_n)$, characterizing conditions on the selection weight functions and data in order that $(\hat{\theta}_n, \mathbb{G}_n)$ 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.


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Peter B. Gilbert. "Large sample theory of maximum likelihood estimates in semiparametric biased sampling models." Ann. Statist. 28 (1) 151 - 194, February 2000.


Published: February 2000
First available in Project Euclid: 14 March 2002

zbMATH: 1106.60302
MathSciNet: MR1762907
Digital Object Identifier: 10.1214/aos/1016120368

Primary: 60G05 , 62F05
Secondary: 62G20 , 62G30

Keywords: Asymptotic theory , choice-based sampling , Clinical trials , Empirical processes , generalized logistic regression , HIV vaccine trial , Nonparametric maximum likelihood , selection bias models , Vardi’s estimator

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 1 • February 2000
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