Estimating a binomial parameter with finite memory

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Abstract
This article treats the asymptotic theory of estimating a binomial parameterpwith time-invariant finite memory. The approach taken to this problem is as follows. A decision rule is a pair(t,a)in whichtfixes the transition function of a finite automaton, andais a vector of estimates ofp. Attention is restricted to automata whose transition functions allow transitions only between adjacent states. Rules(t,a)for whichtsatisfies this restriction are termed tridiagonal. For the class of prior distributions on [0,1] which have continuous density functions, we study the performance of a corresponding class of tridiagonal rules{ (t^{ast},a^{ast}) }relative to quadratic loss functions. These rules display sensitivity to the shape of the prior, and have the advantage that the Bayes estimatea^{ast}(givent^{ast}) is easily computed. Within the class of all tridiagonal rules, a particular rule(t^{ast},a^{ast})is shown, for memory size up to 30, to be locally admissible and minimax as well as locally Bayes with respect to the uniform prior.

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