On the Measurability and Consistency of Maximum Likelihood Estimates for Unimodal Densities

The Annals of Statistics · 1973 · Volume 1 · Issue 5

Abstract

This paper is concerned with maximum likelihood estimates for a large class of families of unimodal densities. The existence of measurable maximum likelihood estimates and the consistency of asymptotic maximum likelihood estimates are proved. By counterexamples it is shown that the conditions which are sufficient for consistency cannot be removed without compensation.

Keywords

28A20, 54A10, 54E45, 62G05, asymptotic maximum likelihood estimates, compact and locally compact metric space, convergence in the mean, existence of measurable estimates, Levy metric, pointwise convergence, strong consistency, the mode of a unimodal density, Unimodal density, upper and lower semicontinuity, weak topoloty and topology induced by the supremum-metric on families of probability measures

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