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**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.

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 |