Large Sample Theory of Empirical Distributions in Biased Sampling Models

Vardi (1985a) introduced an s-sample model for biased sampling, gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator Gnof the common underlying distribution G and discussed numerical methods for calculating the estimator. Here we examine the...

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Veröffentlicht in:	The Annals of Statistics. - Institute of Mathematical Statistics. - 16(1988), 3, Seite 1069-1112
1. Verfasser:	Gill, Richard D. (VerfasserIn)
Weitere Verfasser:	Vardi, Yehuda, Wellner, Jon A.
Format:	Online-Aufsatz
Sprache:	English
Veröffentlicht:	1988
Zugriff auf das übergeordnete Werk:	The Annals of Statistics
Schlagworte:	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 mehr... stratified sampling Vardi's estimator Philosophy Mathematics


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520			\|a Vardi (1985a) introduced an s-sample model for biased sampling, gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator Gnof the common underlying distribution G and discussed numerical methods for calculating the estimator. Here we examine the large sample behavior of the NPMLE Gn, including results on uniform consistency of Gn, convergence of $\sqrt n (\mathbb{G}_n - G)$ to a Gaussian process and asymptotic efficiency of Gnas 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 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.
540			\|a Copyright 1988 Institute of Mathematical Statistics
650		4	\|a Asymptotic theory
650		4	\|a case-control studies
650		4	\|a choice based sampling
650		4	\|a empirical processes
650		4	\|a enriched stratified sampling
650		4	\|a graphs
650		4	\|a lenght-biased sampling
650		4	\|a Neyman allocation
650		4	\|a nonparametric maximum likelihood
650		4	\|a selection bias models
650		4	\|a stratified sampling
650		4	\|a Vardi's estimator
650		4	\|a Philosophy \|x Applied philosophy \|x Philosophy of science \|x Scientific method \|x Research variables \|x Research biases \|x Sampling bias
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Inferential statistics \|x Statistical estimation \|x Estimation methods \|x Estimators
650		4	\|a Mathematics \|x Applied mathematics \|x Analytics \|x Analytical estimating \|x Maximum likelihood estimation
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Descriptive statistics \|x Measures of variability \|x Multivariate statistical analysis \|x Covariance
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Statistical theories \|x Estimation theory \|x Estimation bias
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Inferential statistics \|x Statistical estimation \|x Estimation methods \|x Estimators \|x Maximum likelihood estimators
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Descriptive statistics \|x Statistical distributions \|x Normal distribution curve \|x Sampling distributions
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Statistical theories \|x Asymptotic theory
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Statistical results \|x Statistical properties \|x Identifiability
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Inferential statistics \|x Statistical estimation \|x Estimation methods
655		4	\|a research-article
700	1		\|a Vardi, Yehuda \|e verfasserin \|4 aut
700	1		\|a Wellner, Jon A. \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t The Annals of Statistics \|d Institute of Mathematical Statistics \|g 16(1988), 3, Seite 1069-1112 \|w (DE-627)270129162 \|w (DE-600)1476670-X \|x 00905364 \|7 nnns
773	1	8	\|g volume:16 \|g year:1988 \|g number:3 \|g pages:1069-1112
856	4	0	\|u https://www.jstor.org/stable/2241619 \|3 Volltext
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952			\|d 16 \|j 1988 \|e 3 \|h 1069-1112