Discrete mixture representations of parametric distribution families: geometry and statistics

TL;DR

This paper explores the theoretical existence and properties of discrete mixture representations for parametric families of probability measures, analyzing their geometric and statistical implications such as Fisher information loss and estimation efficiency.

Contribution

It provides new theoretical results on the existence and properties of discrete mixture representations, including their impact on Fisher information and statistical estimation methods.

Findings

Existence results for discrete mixture representations.

Analysis of Fisher information loss in mixture models.

Implications for maximum likelihood estimation of mixing probabilities.

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.