The Annals of Statistics

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

Peter B. Gilbert

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Abstract

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.

Article information

Source
Ann. Statist., Volume 28, Number 1 (2000), 151-194.

Dates
First available in Project Euclid: 14 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1016120368

Digital Object Identifier
doi:10.1214/aos/1016120368

Mathematical Reviews number (MathSciNet)
MR1762907

Zentralblatt MATH identifier
1106.60302

Subjects
Primary: 60G05: Foundations of stochastic processes 62F05: Asymptotic properties of tests
Secondary: 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions

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

Citation

Gilbert, Peter B. Large sample theory of maximum likelihood estimates in semiparametric biased sampling models. Ann. Statist. 28 (2000), no. 1, 151--194. doi:10.1214/aos/1016120368. https://projecteuclid.org/euclid.aos/1016120368


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