Sequential Estimation in Bernoulli Trials

Abstract

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 $p$.