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 σ²E(y x) ≤≤ σ²y, 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 θ, the function E(t u), which is a function of u only, is an unbiased estimate for θ 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.