Estimation of Parameter n of the Binomial Distribution

Abstract

Let S₁, …, S_k be a random sample of size k from a binomial distribution with parameters n and θ, the success probability. We are interested in estimating n when both θ and n are unknown. We use a Bayesian approach to estimating n. Even though, from the Bayesian viewpoint, the number of trials is a discrete random variable N, we take a continuous prior distribution for N. A justification for using this continuous prior distribution is given. Assuming the quadratic loss function, the mean of the posterior distribution of N is the Bayes estimator of n. The Bayes estimator does not possess a closed form. It is evaluated by using the Laguerre-Gauss quadrature.

A Monte Carlo study of the Bayes, the stable versions of the method of moments and the maximum likelihood (Olkin, Petkau and Zidek, 1981) and the Carroll-Lombard (1985) estimators are made. The numerical work indicates that the Bayes estimator is a stable estimator and, in some cases, is superior to other n estimators in terms of the mean squared error and no single estimator dominates all others in all cases.

Keywords:

• Bayes
• Estimator
• Binomial distribution
• Laguerre-Gauss quadrature