Asymptotic normality of unbiased estimators in the case of an absolutely continuous multiparameter exponential family

Abstract

Sufficient conditions are obtained for the asymptotic normality of unbiased estimators for the distribution density of a random vector with independent identically distributed components and for the linear integral functional of this density in the case of an exponential family of distributions. To analyze the limit behavior of unbiased estimators, multidimensional local limit theorems for density are used. Under certain regularity conditions, the limit law variance is demonstrated to coincide with the Cramer–Rao lower bound.