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A note on Hammersley’s inequality for estimating the normal integer mean. (English) Zbl 1022.62018

Summary: Let \(X_{1}, X_{2},\dotsc,X_{n}\) be a random sample from a normal \(N(\theta,\sigma^2)\) distribution with an unknown mean \(\theta = 0,\pm 1, \pm 2,\ldots\). J. M. Hammersley [J. R. Stat. Soc., Ser. B 12, 192-240 (1950; Zbl 0040.22202)] proposed the maximum likelihood estimator (MLE) \(d =[\overline{X}_n]\), nearest integer to the sample mean, as an unbiased estimator of \(\theta\) and extended the Cramér-Rao inequality. The Hammersley lower bound for the variance of any unbiased estimator of \(\theta\) is significantly improved, and the asymptotic (as \(n\rightarrow\infty\)) 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.

MSC:

62F10 Point estimation
62F12 Asymptotic properties of parametric estimators

Citations:

Zbl 0040.22202
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