Conditional Expectation and Unbiased Sequential Estimation

It is shown that E f (x) E (y x) = E (fy) whenever E (fy) is finite, and that ²E (y x) ²y, where E (y 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, the function E (t u), which is a function of u only, is an unbiased estimate for 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.