International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2003 / Article

Open Access

Volume 2003 |Article ID 314030 | https://doi.org/10.1155/S016117120320822X

Rasul A. Khan, "A note on Hammersley's inequality for estimating the normal integer mean", International Journal of Mathematics and Mathematical Sciences, vol. 2003, Article ID 314030, 10 pages, 2003. https://doi.org/10.1155/S016117120320822X

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

Received07 Aug 2002

Abstract

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=[X¯n], 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.

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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