Admissible Estimation of the Binomial Parameter $n$

Abstract

Suppose that $X$ has a binomial distribution $B(n, p)$, with known $p \in (0, 1)$ and unknown $n \in \{1, 2, \cdots\}$. A natural estimator for $n$ is given by $T(0) = 1, T(x) = x/p, x = 1, 2, \cdots$. This estimator is shown to be inadmissible under quadratic loss. It is shown that modifying $T(0)$ to $T(0) = -(1 - p)/(p \ln p)$ results in an admissible estimator. For $p \geq \frac{1}{2}$ it is further shown that this is the only admissible modification of $T(0)$. A partial result is also obtained for $p < \frac{1}{2}$.