Statistical analysis of mixtures and the empirical probability measure
type: Item
instance of: scholarly article
title: Statistical analysis of mixtures and the empirical probability measure
zbMATH Open document ID: 0917.62028
DOI: 10.1023/A:1005824513112
author: Philippe Barbe
published in: Acta Applicandae Mathematicae
publication date: 1999-06-21T00:00:00Z
review text: The paper concerns the problem of estimating mixture probability measures in an abstract setting. The treated problem can be formulated as follows: Let \(\mathcal P=\{P_{\theta}; \theta \in \Theta\}\) be a family of probability measures defined on a separable metric space \(\mathcal X\), where the parameter space \(\Theta\) is also a separable metric space. The mixture is defined as \(P_{\mu}=\int P_{\theta}d\mu_{\theta},\) where \(\mu\) is a probability measure on \(\Theta\). Given a sample of i.i.d. random variables \(X_1,..,X_n\) with a common distribution \(P_{\theta_0}\) one wants to estimate \(\mu_0\) if the family \(\mathcal P\) is known and if \(\mu\in \mathcal M \) with \(\mathcal M\) known. A partial review of the literature on mixture models is presented. New results are proved and compared with the already published ones. Five topics are considered: identifiability, classes of mixtures, maximum likelihood estimation and other methods of estimation, and efficiency of the estimators. The paper is divided into 10 sections and an Appendix. Section 1 is introductory, the consistency of minimum distance estimators is proved in Section 2. Section 3 contains general results on the asymptotic behavior of the distance of the estimators \(P_{\mu_n}\) of \(P_{\mu_0}\) and the empirical measure \(P_n\). A necessary and sufficient condition for \(P_{\mu_n}\) to behave asymptotically as \(P_n\) is given. Section 4 shows how results on \(P_{\mu_n}\) are related to \(\mu_n\). Section 5 shows that when treating the minimum distance estimators identifiability is not needed. Basic properties of functions associated with mixing distributions of the limiting distribution of the distance between \(P_{\mu_n}\) and \(P_n\) are studied. The theory developed in Sections 2-6 is illustrated by a number of examples in Sections 7 and 9. Representations of the inverse of an operator canonically associated with the family \(\mathcal P\) are studied. Remarks and open problems are formula…
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 41A35
Mathematics Subject Classification ID: 45P05
zbMATH DE Number: 1139550
zbMATH Keywords: mixture models
zbMATH Keywords: empirical processes
zbMATH Keywords: approximation by linear operators
zbMATH Keywords: integral transformations
reviewed by: Marie Hušková
MaRDI profile type: Publication
full work available at URL: a:1005824513112
OpenAlex ID: W157970812
description: scientific article; zbMATH DE number 1139550