On Pitman domination

Date
1993
Authors
Yoo, Seongmo
Major Professor
Advisor
Herbert T. David
Committee Member
Journal Title
Journal ISSN
Volume Title
Publisher
Altmetrics
Authors
Research Projects
Organizational Units
Statistics
Organizational Unit
Journal Issue
Series
Department
Statistics
Abstract

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.

Comments
Description
Keywords
Citation
Source