Mixtures and Products of Dominated Experiments

Abstract

It is shown, using a theorem of Choquet, that any separable experiment is a mixture of experiments admitting boundedly complete and sufficient statistics. The experiments possessing these properties are precisely the experiments which are extremal with respect to mixtures. Dominated models for independent observations $X_1, \cdots, X_n$ admitting boundedly (or $L_p$) complete and sufficient statistics, are considered. It is shown that a subset--say $X_1, \cdots, X_m$ where $m < n$--has the same property provided a regularity condition is satisfied. This condition is automatically satisfied when the observations are identically distributed. In the bounded complete case the proof uses the fact that products of experiments are distributive w.r.t. mixtures. More involved arguments are needed for $L_p$ completeness.