Statistical analysis of mixtures and the empirical probability measure
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 formulated in Section 10. The paper presents a useful collection of theoretical results on mixture models.
scientific article; zbMATH DE number 1139550