Abstract
We consider the sequential estimation of p, the probability of success in an infinite sequence of Bernoulli trials, when the loss incurred in stopping after n trials and estimating p by some function δn of the first n outcomes is taken to be [Formula: see text] The loss due to error of estimation is thus the symmetrized relative squared error, which is appropriate in applications when p may be near 0 or 1.
We begin by finding a heuristic procedure for sequentially determining the sample size n which, with the usual terminal estimator of p, performs well for any fixed 0 < p < 1 as the cost per observation c → 0. For any fixed c > 0, however, this procedure is poor as p → 0 or 1. To remedy this defect, the uniform prior on p is introduced. The corresponding Bayes procedure is found and is shown to have a Bayes risk ∼2π√c as c → 0.
Keywords: statistics, sequential estimation, optimal Bayes
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