A Note on Hammersley’s Inequality for Estimating the Normal Integer Mean

Let X1,X2, . . . ,Xn be a random sample from a normal N(θ,σ2) distribution with an unknown mean θ = 0,±1,±2, . . . . Hammersley (1950) proposed the maximum likelihood estimator (MLE) d = [Xn], nearest integer to the sample mean, as an unbiased estimator of θ and extended the Cramér-Rao inequality. The Hammersley lower bound for the variance of any unbiased estimator of θ is significantly improved, and the asymptotic (as n → ∞) limit of Fraser-Guttman-Bhattacharyya bounds is also determined. A limiting property of a suitable distance is used to give some plausible explanations why such bounds cannot be attained. An almost uniformly minimum variance unbiased (UMVU) like property of d is exhibited.