Statistica Sinica 4(1994), 679-692

ON FISHER'S BOUND FOR STABLE ESTIMATORS

WITH EXTENSION TO THE CASE OF HILBERT

PARAMETER SPACE

Zhongguo Zheng and Kai-Tai Fang

Peking University and Hong Kong Baptist College

Abstract: Fisher's bound for asymptotic variances has been shown to hold almost everywhere for several classes of estimators in LeCam (1953), Bahadur (1964) and Wong (1992). Wong's result applies to estimators with arbitrary asymptotic distributions, provided a certain continuity condition is satisfied. In this paper we improve on Wong's result by removing the continuity condition. We also generalize the result to the case where the parameter space is separable Hilbert.

Key words and phrases: Concentration probability, Fisher's bound of estimator, asy- mptotic efficiency.