On Pitman domination

Abstract

<p>According to Pitman (1937), an estimator X is closer than an estimator Y to a scalar parameter [theta] (or, in the terminology used below, X Pitman-dominates Y) if Pr[subscript][theta]( ǁ X - [theta] ǁ 1/2, ∀[theta].This criterion is now called the Pitman Closeness Criterion (PCC). Pitman suggested that median-unbiased estimators derived from sufficient statistics are well suited to PCC, and noted that FCC is intransitive;After Pitman gave the "comparison theorem" for identifying classes of estimators Pitman-dominated by median-unbiased estimators derived from sufficient statistics, Ghosh and Sen (1989) and Nayak (1990) showed that a median-unbiased estimator is best equivariant in the Pitman sense. These investigations are in a sense supportive of Pitman's idea;Following a different line of research based on certain shrinkage constructions, Salem and David (1973) constructed a class of continuous increasing functions of a median-unbiased estimator Pitman-dominating the sample mean for estimating the mean [theta] of a normal density with known variance (see also Efron (1975) for an example in a similar vein). David and Salem (1991) extended the result of Salem and David (1973) to the case of a single observation from any symmetric density, and also constructed intransitive triples of estimators of a Laplace location parameter, each member of the triple Pitman-dominating the single observation. This direction of research is less supportive of Pitman's idea;We generalize the approach of David and Salem (1991). A number of parametric situations are considered, including some considered by Pitman. In each case, a class of continuous not necessarily increasing functions of a median-unbiased or otherwise natural estimator derived from sufficient statistics is considered, each member of the class Pitman-dominating the estimator itself. Special attention is given to Pitman domination for location-scale families. Finally, we construct Pitman-intransitive triples of estimators based on the earlier results on shrinkage and equivariant estimators.</p>