Statistics & Probability Letters 19 (1994) 131-136 North-Holland 27 January 1994 The Pitman comparison of unbiased linear estimators Robert L. Fountain zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Portland State University, Portland, OR, USA Jerome P. Keating The University of Texas at San Antonio, USA Received April 1992 Revised February 1993 Abstract: The canonical form for the comparison of certain linear estimators using Pitman’s Measure of Closeness is generalized to the class of all linear estimators. Under the assumption of normality, the equivalence of Pitman-closest linear unbiased estimators and best linear unbiased estimators is shown. A sufficient condition is given for which the BLUE will be Pitman-closer than the best linear equivariant estimator (BLEE). Keywords: Pitman’s measure of closeness; order statistics; best linear unbiased estimators; best linear equivariant estimators. 1. Introduction Pitman’s Measure of Closeness (PMC) has been the subject of many recent investigations. As a pairwise comparison of estimators of a common parameter, it was introduced by Pitman (1937). Rao (1981) and Keating and Mason (1985a, 1985b) helped to resurrect the criterion as an alternative to classical standards such as the mean squared error. Fountain, Keating and Rao (1991) used PMC to re-evaluate the performance of various modifications of minimum x2 and maximum like- lihood estimators. Ghosh and Sen (1989) and Nayak (1990) have provided estimators which are ‘best’ in the Pitman sense, within restricted classes. Keating (1991) has unified many of the known univariate results within a topological set- Correspondence to: Robert L. Fountain, Department of Math- ematical Sciences, Portland State University, Portland, OR 97207-0751, USA. ting. But the calculation of PMC in the general case remain: a difficult problem. Letting 8, and g2 be two estimators of the parameter 8, the PMC of e^, relative to e^, in estimating 0 under a loss function L,(. > is Most commonly, &(I?) = I e^ - 13 I PMC is invariant over powers of tion. . Notice that this loss func- Rao, Keating and Mason (1986 1) have given a geometric approach to the calculation of PMC. Mason, Keating, Sen and Blaylock (1990) have further discussed the computation, with applica- tions to ridge regression estimators. In the case of linear estimators, Fountain (1991) gave a canoni- cal form for PMC based on the spectral decom- position of a matrix formed from the coefficient vectors. Peddada and Khattree (1991) provided some conditions for preference of linear estima- 0167-7152/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0167-7152(93)E0093-9 131