Discrete mixture representations of parametric distribution families: Geometry and statistics

Abstract

We investigate existence and properties of discrete mixture representations Pθ = ∑i∈Ewθ(i) Qi for a given family Pθ, θ ∈Θ, of probability measures. The noncentral chi-squared distributions provide a classi-cal example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher in-formation, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.