International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 34, Pages 2147-2156
doi:10.1155/S016117120320822X
A note on Hammersley's inequality for estimating the normal
integer mean
Department of Mathematics, Cleveland State University, 2121 Euclid Avenue, RT 1515, Cleveland 44115, OH, USA
Received 7 August 2002
Copyright © 2003 Rasul A. Khan. 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.
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.