The Consistency of Certain Sequential Estimators

Abstract

The results described here have their roots in two areas, for in a certain sense we combine on the one hand the work of Girshick, Mosteller and Savage [5] and Wolfowitz [11] and [12] on sequential estimation of the binomial parameter, and on the other the result of Hoeffding [7] concerning the consistency of $U$-statistics. The link between the two is the Blackwell [2] procedure for obtaining another (better) estimator from a given one by taking expectations conditional on a sufficient statistic. The main result is that if from a given estimator $T$ of $\theta = ET$ we construct new estimators by the Blackwell procedure corresponding to a sequence of stopping-rules $N_i$, then this sequence of estimators is consistent provided $N_i$ tends to infinity in probability; in fact it has also to be assumed that the $N_i$ have a certain structural property.