Weak Convergence of the Rao-Blackwell Estimator of a Distribution Function

Abstract

Under the condition that the minimal sufficient statistics are transitive, the sequence of Rao-Blackwell estimators of distribution function has been shown to form a reverse martingale sequence. Weak convergence of the corresponding empirical process to a Gaussian process has been established by assuming that the sufficient statistics are $U$-statistics and utilizing certain results on the convergence of conditional expectations of functions of $U$-statistics along with the functional central limit theorems for (reverse) martingales by Loynes (1970) and Brown (1971).