Conditional Expectation and Unbiased Sequential Estimation

It is shown that $E\lbrack f(x) E(y \mid x)\rbrack = E(fy)$ whenever $E(fy)$ is finite, and that $\sigma^2E(y \mid x) \leq\le \sigma^2y$, where $E(y \mid x)$ denotes the conditional expectation of $y$ with respect to $x$. These results imply that whenever there is a sufficient statistic $u$ and an unbiased estimate $t$, not a function of $u$ only, for a parameter $\theta$, the function $E(t \mid u)$, which is a function of $u$ only, is an unbiased estimate for $\theta$ with a variance smaller than that of $t$. A sequential unbiased estimate for a parameter is obtained, such that when the sequential test terminates after $i$ observations, the estimate is a function of a sufficient statistic for the parameter with respect to these observations. A special case of this estimate is that obtained by Girshick, Mosteller, and Savage [4] for the parameter of a binomial distribution.