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

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  • Rasul A. Khan

Abstract

Let X 1 , X 2 , … , X n be a random sample from a normal N ( θ , σ 2 ) 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.

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  • Rasul A. Khan, 2003. "A note on Hammersley's inequality for estimating the normal integer mean," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-10, January.
    • Handle: RePEc:hin:jijmms:314030
    DOI: 10.1155/S016117120320822X
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    Cited by:

    1. Luati, Alessandra & Novelli, Marco, 2020. "The Hammersley–Chapman–Robbins inequality for repeatedly monitored quantum system," Statistics & Probability Letters, Elsevier, vol. 165(C).

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