Unbiased Estimates for Certain Binomial Sampling Problems with Applications

5. Conclusion

We would like to call attention to a few problems raised by but not solved in this paper: 1) find a necessary and sufficient condition that $\hat p$ be the unique unbiased estimate for p; 2) suggest criteria for selecting one unbiased estimate when more than one is possible; 3) evaluate the variance of $\hat p$.

In this connection, in a forthcoming paper by M.A. Girshick, it will be shown for certain regions, for example for those of the sequential probability ratio test, that the variance of $\hat p(\alpha )$,

$$ \sigma _{\hat p}^2 \geqslant pq/E(x + y),$$

where E(x + y) is the expected number of observations required to reach a boundary point.

This paper was originally written by Mosteller and Savage. A communication from M.A. Girshick revealed that he had independently discovered for the sequential probability ratio test the estimate $\hat p(\alpha )$ given here and demonstrated its uniqueness. For purposes of publication it seemed appropriate to present the results in a single paper.