Estimating the integer mean of a normal model related to binomial distribution

Abstract

A problem for estimating the number of trials $n$ in the binomial distribution $B(n,p)$, is revisited by considering the large sample model $N(\mu ,c\mu )$ and the associated maximum likelihood estimator (MLE) and some sequential procedures. Asymptotic properties of the MLE of $n$ via the normal model $N(\mu ,c\mu )$ are briefly described. Beyond the asymptotic properties, our main focus is on the sequential estimation of $n$. Let $X_1,X_2,\dots ,X_m,\dots$ be iid $N(\mu ,c\mu )$$(c>0)$ random variables with an unknown mean $\mu =1,2,\dots$ and variance $c\mu$, where $c$ is known. The sequential estimation of $\mu$ is explored by a method initiated by Robbins (1970) and further pursued by Khan (1973). Various properties of the procedure including the error probability and the expected sample size are determined. An asymptotic optimality of the procedure is given. Sequential interval estimation and point estimation are also briefly discussed.