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
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.