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Let X1,X2,Ã¢Â€Â¦,Xn be a random sample from a normal N(ÃŽÂ¸,ÃÂƒ2) distribution with an unknown mean ÃŽÂ¸=0,Ã‚Â±1,Ã‚Â±2,Ã¢Â€Â¦. Hammersley (1950) proposed the maximum likelihood estimator (MLE) d=[XÃ‚Â¯n], nearest integer to the sample mean, as an unbiased estimator of ÃŽÂ¸ and extended the CramÃƒÂ©r-Rao inequality. The Hammersley lower bound for the variance of any unbiased estimator of ÃŽÂ¸ is significantly improved, and the asymptotic (as nÃ¢Â†Â’Ã¢ÂˆÂž) 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

A note on Hammersley's inequality for estimating the normal integer mean