Improving on the MLE of $p$ for a binomial $(n,p)$ when $p$ is around $\frac 12$
François Perron · Lecture notes-monograph series · 2003
We consider the problem of estimating the parameter p of a binomial (n,p) distribution when p lies in a symmetric interval of length m/y/n, m < y/n.We establish sufficient conditions for the domination of the maximum likelihood estimator with quadratic loss.We suggest three other estimators for the estimation of p.The first two dominate the maximum likelihood estimator.The first one comes from Moors (1985) and corresponds to the bayesian estimator with respect to the symmetric prior concentrated on the end points if and only if TO < 1 when n is odd or m < y/n/(n -1) when n is even.The second estimator comes from Charras and van Eeden (1991); it is in fact the maximum likelihood estimator for the problem where m is replaced by TOO, 0 < mo < TO.We give an algorithm for the selection of TOO-The third is the Bayes estimator with respect to the prior having a density proportional to (p(l -p)) .This estimator dominates the maximum likelihood estimator for some values of (n, TO) but not for all of them.We give simple sufficient conditions for the domination of the Bayes estimator over the maximum likelihood estimator.It is clear that the maximum likelihood estimator is inappropriate when either n or TO is small.When n is large, all of the estimators have approximately the same behaviour except for the last.Numerical evaluations illustrate our comments.