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 $\theta$ 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 $g(\theta)$ in the sense of Bahadur, for virtually all functions $g$ 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.