Bounds for robust maximum likelihood and posterior consistency in compound mixture state experiments

Uniform bounds on rates of L1-consistency for the empiric mean of state sequences in a family of probability models (compound with finite-mixture-state component) are obtained for MLEs (Section sec2) and posterior means for quasi-uniform hyperpriors (Section sec3), both determined in the iid mixture (empirical Bayes) sub-models. Qualitative aspects of results of this type were described by Robbins (1951). Application to the Gilliland and Hannan (1974/86) restricted-risk-finite-state-component compound decision problem (Section sec4) yields uniform bounds on rates of asymptotic regret of Bayes solutions therein (with extension to mixture-state by expectation), giving strong affirmation to an asymptotic form of a Robbins (1951) conjecture. The general extension to mixture-state components (Remark rem4.1) strengthens much of the existing compound literature.