Discrete mixture representations of parametric distribution families: geometry and statistics

Abstract

We investigate existence and properties of discrete mixture representations $P_{\theta }={\sum }_{i\in E}{w}_{\theta }(i)Q_i$ for a given family $P_{\theta }$, $\theta \in \Theta$, of probability measures. The noncentral chi-squared distributions provide a classical example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher information, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.