Pitman Closeness in Classes of General Pre-Test Estimators and Regression Estimators

ABSTRACT

For the task of “optimal” estimation of a (vector) parameter we consider two Pitman closeness type criterions. The first one uses the generalized Pitman closeness measure (see e.g., Saleh and Sen, 1991). The second one has been introduced by Ghosh and Sen (1991) and independently by Nayak (1991); we call it the average generalized closeness criterion. Pitman closeness type comparisons suffer from non transitivity in general. Nevertheless, within very general classes of pre-test estimators (PTE) we prove transitivity and other useful properties for the two criterions. Further, we restrict ourselves to linear regression models. We compare the ordinary least squares estimator with the restricted least squares estimator employing one restriction and discuss existence of best PTE corresponding to these models under the criterions mentioned above.

Keywords:

• Linear regression
• Pitman closeness criterion
• Pre-test estimator

Mathematics Subject Classification:

• 62F10
• 62H12
• 62F15
• 62J05

Acknowledgment

The author was supported by the Project MSM 4977751301 of Ministry of Education of the Czech Republic.