On Some Properties of Hammersley's Estimator of an Integer Mean

Abstract

Let $X_1, \cdots, X_n$ be i.i.d. $N(i, 1), i = 0, \pm 1, \pm 2,\cdots$. Hammersley [2] proposed $\lbrack\bar{X}_n\rbrack$, the nearest integer to the sample mean, as an estimator of $i$. It is proved that $d$ is minimax and admissible relative to zero-one loss. However, it is shown that relative to squared error loss, the estimator is neither admissible nor minimax.