Large Sample Theory of Empirical Distributions in Biased Sampling Models

Gill, Richard D. ; Vardi, Yehuda ; Wellner, Jon A.

Ann. Statist., Tome 16 (1988) no. 1, p. 1069-1112 / Harvested from Project Euclid

Résumé

Vardi (1985a) 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 the estimator. Here we examine the large sample behavior of the NPMLE $\mathbb{G}_n$, 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 $\mathbb{G}_n$ as an estimator of $G$. The proofs are based upon recent results for empirical processes indexed by sets and functions and convexity arguments. We also give a careful proof of identifiability of the underlying distribution $G$ under connectedness of a certain graph $\mathbf{G}$. 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.

Détail

Publié le : 1988-09-14
Classification:  Asymptotic theory,  case-control studies,  choice based sampling,  empirical processes,  enriched stratified sampling,  graphs,  lenght-biased sampling,  Neyman allocation,  nonparametric maximum likelihood,  selection bias models,  stratified sampling,  Vardi's estimator,  62G05,  60F05,  62G30,  60G44