Minimax Estimation of a Constrained Binomial Proportion p When | p − 1 / 2 | is Small

We consider the problem of estimating the parameter p of a Binomial(n, p) distribution when p lies in the symmetric interval about 1/2 of the form [a, 1 − a], with a ∈ (0, 1/2). For a class of loss functions, which includes the important cases of squared error and information-normalized losses, we investigate conditions for which the Bayes estimator, δBU , with respect to a symmetric prior concentrated on the end points of the parameter space is minimax. Our conditions are of the form 1−2a ≤ c(n) with c(n) = O(n−1/2), and various analytical evaluations, lower and upper bounds, and numerical evaluations are given for c(n). For instance, the simple condition 1 − 2a ≤ 1/ √ 2n guarantees, for all n ≥ 1, the minimaxity of δBU under both squared error and information-normalized losses. AMS (2000) subject classification. 62F10, 62F15, 62F30 62F99.