Conditional Expectation and the Efficiency of Estimates

Abstract

A probability density function, $f(x; \theta)$, is considered for which there exists a sufficient statistic. It is shown, under certain regularity conditions on the family of distributions and on the class of estimates, that if there exists an unbiased sufficient estimate of $\theta$, it will be unique. This result is used to show that when the regularity conditions are satisfied, the method of Blackwell for improving an unbiased estimate of $\theta$ merely yields a natural estimate.