Monotone Convergence of Binomial Probabilities with an Application to Maximum Likelihood Estimation

Abstract

It has been pointed out in the literature that the maximum likelihood (ML) estimator may be misleading in the presence of prior information. Many of these examples assume extreme sizes: one or infinity. In the present paper an example is considered where sample size may be any odd positive integer. This example is an amplification of the one given by Lehmann (1949) where he considers estimation of the probability of "success" $p$, based on a single observation of a Bernoulli random variable $X$. He states that with the prior information $\frac{1}{3} \leqq p \leqq \frac{2}{3}$ the ML estimator has uniformly larger expected squared error than any estimator $\delta(X)$ which is symmetric about $\frac{1}{2}$ and is such that $\frac{1}{3} \leqq \delta(0) \leqq \frac{1}{2} \leqq \delta(1) \leqq \frac{2}{3}$. In particular, the ML estimator is uniformly worse than the trivial estimator $\delta(X) \equiv \frac{1}{2}$. A natural question arises: does the same phenomenon occur for larger samples? In the following it has been shown that with $(2n + 1)$ observations if $p$ is known to be in a small interval around $\frac{1}{2}$ then the trivial estimator is uniformly better than the ML estimator [now] based on $(2n + 1)$ observations. The interval having this property shrinks as $n$ becomes large. The proof is based on a monotone convergence of certain binomial probabilities which itself may be of some interest.