Existence and Consistency of Modified Minimum Contrast Estimates

Abstract

It is the purpose of this paper to explore the efficiency of a modified definition for maximum likelihood estimates which depends on the whole equivalence class of densities only and not--as in the classical case--on the particular choice of versions. We prove the existence of measurable maximum likelihood estimates in the new sense for compact metrizable families of probability measures without any continuity assumption for the densities. For appropriate families of probability measures the modified asymptotic maximum likelihood estimates are exactly the strongly consistent estimates. The paper uses Huber's concept of minimum contrast estimates which covers maximum likelihood estimates as a special case.