Sankhya: The Indian Journal of Statistics

2005, Volume 67, Pt. 3, 526--537

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

Eric Marchand, University of New Brunswick and Universit\'e de Sherbrooke, Canada
Fran\c cois Perron and Rokhaya Gueye, Universit\'e de Montr\'eal, Canada

SUMMARY. 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 $a \in (0,1/2)$. 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, $\delta_{BU}$, with respect to a symmetric prior concentrated on the end points of the parameter space is minimax.  Our conditions are of the form $1-2a \leq c(n)$ 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 $1-2a \leq {1}/{\sqrt{2n}}$ guarantees, for all $n \geq 1$, the minimaxity of $\delta_{BU}$ under both squared error and information-normalized losses.

AMS (1991) subject classification. 62F10, 62F15, 62F30 62F99.

Key words and phrases. Minimax estimation, restricted parameter space, binomial distribution, squared error loss, information-normalized loss.

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