The Annals of Mathematical Statistics
- Ann. Math. Statist.
- Volume 40, Number 2 (1969), 568-574.
The Consistency of Certain Sequential Estimators
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  and Wolfowitz  and  on sequential estimation of the binomial parameter, and on the other the result of Hoeffding  concerning the consistency of $U$-statistics. The link between the two is the Blackwell  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.
Ann. Math. Statist., Volume 40, Number 2 (1969), 568-574.
First available in Project Euclid: 27 April 2007
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Loynes, R. M. The Consistency of Certain Sequential Estimators. Ann. Math. Statist. 40 (1969), no. 2, 568--574. doi:10.1214/aoms/1177697724. https://projecteuclid.org/euclid.aoms/1177697724