The Asymptotic Inadmissibility of the Sample Distribution Function

Abstract

Given a sample of size $n$, a continuous estimator for a distribution $F$ (based on Pyke's modified sample distribution) is shown to have the property that its expected squared error, for almost all $x$ in the positive sample space of $F$, is no larger than that of the sample distribution function given $F$ and $n$ sufficiently large. Letting risk be given by the expected squared error integrated with respect to $F$, it is shown that this estimator dominates both the sample distribution and the other best invariant estimator found by Aggarwal, given $F$ and $n$ sufficiently large. Other common estimators cannot serve in this dominating role. Explicit calculation of risk is made when $F$ is the uniform distribution. In this case the estimator strictly dominates the sample distribution for all $n \geqq 1$.