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March, 1977 Sequential Estimation in Bernoulli Trials
Paul Cabilio
Ann. Statist. 5(2): 342-356 (March, 1977). DOI: 10.1214/aos/1176343799


The sequential estimation of $p$, the probability of success in a sequence of Bernoulli trials, is considered for the case where loss is taken to be symmetrized relative squared error of estimation, plus a fixed cost $c$ per observation. Using $s_n/n$ as a terminal estimator of $p$, where $s_n$ is the number of successes in $n$ trials, a heuristic rule is derived and shown to perform well for any fixed $0 < p < 1$ as $c \rightarrow 0$. However for any fixed $c > 0$, this rule performs badly for $p$ close to 0 or 1. To overcome this difficulty a uniform prior on $p$ is introduced, and the optimal Bayes procedure is shown to exist and to have bounded sample size. The optimal Bayes risk is shown to be $\sim 2\pi c^{\frac{1}{2}}$ as $c \rightarrow 0$, and is computed for various values of $c$, along with the expected loss for various values of $p$.


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Paul Cabilio. "Sequential Estimation in Bernoulli Trials." Ann. Statist. 5 (2) 342 - 356, March, 1977.


Published: March, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0363.62064
MathSciNet: MR433761
Digital Object Identifier: 10.1214/aos/1176343799

Primary: 62L12
Secondary: 62L15

Keywords: Asymptotic risk , Bayes rules , Bernoulli trials , Monotone case , Optimal stopping , relative squared error loss , sequential estimation , Stopping rules

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 2 • March, 1977
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