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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

MSC:
 62F10 Point estimation