Statistica Sinica 11(2001), 479-498

ON BAHADUR EFFICIENCY AND MAXIMUM LIKELIHOOD

ESTIMATION IN GENERAL PARAMETER SPACES

Xiaotong Shen

The Ohio State University

Abstract: The paper studies large deviations of maximum likelihood and related estimates in the case of i.i.d. observations with distribution determined by a parameter taking values in a general metric space. The main theorems provide sufficient conditions under which an approximate sieve maximum likelihood estimate is an asymptotically locally optimal estimate of in the sense of Bahadur, for virtually all functions of interest. These conditions are illustrated by application to several parametric, nonparametric, and semiparametric examples.

Key words and phrases: Asymptotic optimality, Bahadur bound, large deviations, maximum likelihood estimation, nonparametric and semiparametric models, the method of sieves.