Minimax Stopping Rules when the Underlying Distribution is Uniform

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Abstract
An invariance-based method of obtaining the minimax stopping rule when sampling from an unknown uniform distribution is presented and applied to two problems, maximizing the probability of selecting the smallest observation and minimizing the expected quantile of the observation selected. In the first problem, the minimax rules use only the relative ranks of the observations; in the second problem they are shown to achieve asymptotic risk (3 + 2 √2)1/√2/n ≈ 3.4780/n, which is intermediate between the values ≈ 3.8695/n for the best rules based on relative ranks and 2/n when the distribution is known. Except for a few small values of the sample size, n, the minimax rules are the formal Bayes rules with respect to an improper a priori “density” whose a posteriori density given the first two observations is proper.

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