Admissible Estimation of the Binomial Parameter n

Suppose that X has a binomial distribution B(n, p), with known p ∈ (0, 1) and unknown n ∈ {1, 2, ⋯}. A natural estimator for n is given by T(0) = 1, T(x) = x/p, x = 1, 2, ⋯. 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 ≥ 1/2 it is further shown that this is the only admissible modification of T(0). A partial result is also obtained for $p < \frac{1}{2}$.