On Bahadur efficiency and maximum likelihood estimation in general parameter spaces

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 θ 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(θ) 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.