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Pitman’s measure of closeness. (English) Zbl 0581.62028
Let \({\hat \theta}{}_ 1,{\hat \theta}_ 2,...,{\hat \theta}_ k\) be k real-valued estimators of a real parameter \(\theta\). Denote the absolute error loss of the i-th estimator by \(L_ i\) where \(L_ i=| {\hat \theta}_ i-\theta |\), \(i=1,2,...,k\). For \(k=2\) E. J. G. Pitman [Proc. Camb. Philos. Soc. 33, 212-222 (1937; Zbl 0016.36404)] introduced \(\Pr (L_ 1<L_ 2)\) as a pairwise measure of closeness for the estimators \({\hat \theta}{}_ 1\) and \({\hat \theta}{}_ 2.\)
In the presence of \(k>2\) estimators of \(\theta\), the authors obtain a generalization of Pitman’s measure of closeness by using a closest approach partition. A procedure for the calculation of this simultaneous Pitman measure is given for a specific class of estimators and illustrated by examination of a one-parameter exponential failure model. Finally, Pitman’s measure is shown to provide information that cannot be ignored when several estimators are being compared.
Reviewer: J.Melamed

62F10 Point estimation