## Large sample theory of empirical distributions in biased sampling models.(English)Zbl 0668.62024

Authors’ abstract: The second author [Ann. Stat. 13, 178-205 (1985; Zbl 0578.62047)] introduced an s-sample model for biased sampling, gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator $${\mathbb{G}}_ n$$ of the common underlying distribution G and discussed numerical methods for calculating $${\mathbb{G}}_ n.$$
The present paper examines the large sample behavior of $${\mathbb{G}}_ n$$, including results on uniform consistency of $${\mathbb{G}}_ n$$, convergence of $$\sqrt{n}({\mathbb{G}}_ n-G)$$ to a Gaussian process and asymptotic efficiency of $${\mathbb{G}}_ n$$. The proofs are based upon recent results for empirical processes indexed by sets and functions and convexity arguments. A careful proof of identifiability of the underlying distribution G under connectedness of a certain graph is presented.
Examples and applications include length-biased sampling, stratified sampling, enriched stratified sampling, choice-based sampling in econometrics and case-control studies in biostatistics. A final section discusses design issues and further problems.
Reviewer: P.Gänßler

### MSC:

 62G05 Nonparametric estimation 62G30 Order statistics; empirical distribution functions 62G20 Asymptotic properties of nonparametric inference 60F05 Central limit and other weak theorems 60G44 Martingales with continuous parameter

Zbl 0578.62047
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