On the measurability and consistency of minimum contrast estimates

Abstract

The concept of minimum contrast (m.c.) estimates used in this paper covers maximum likelihood (m.l.) estimates as a special case. Section 1 contains sufficient conditions for the existence of measurable m.c. estimates and for their consistency. The application of these results to m.l. estimates (section 2) yields the existence of m.l. estimates for families ofp-measures (probability measures) which are compact metric or locally compact with countable base, admitting upper semicontinuous densities, whereas the classical results refer to continuous densities. This generalization is insofar of interest as upper semicontinuous versions of the densities exist whenever the densities areμ-upper semicontinuous (whereasμ-continuity does not, in general, entail the existence of continuous versions). Under appropriate regularity conditions, consistency of asymptotic maximum likelihood estimates is proven for compact (and also locally compact) separable metric families ofp-measures with upper semicontinuous densities and for arbitrary families having uniformly continuous densities with respect to the uniformity of vague convergence. The conditions sufficient for consistency are shown “indispensable” by counterexamples. Section 3 contains auxiliary results. Besides their relevance for sections 1 and 2, some of them may also be of interest in themselves, e.g. Theorem (3.4) on the selection of semicontinuous functions from semicontinuous equivalence classes.