Statistics Papers

Document Type

Journal Article

Date of this Version

1990

Publication Source

Annals of Statistics

Volume

18

Issue

2

Start Page

471

Last Page

493

DOI

10.1214/aos/1176347602

Abstract

Consider a multiple linear regression in which Yi, i=1,⋯, n, are independent normal variables with variance σ2 and E (Yi) = α+V′iβ, where Vi Rr and β Rr. Let α^ denote the usual least squares estimator of α. Suppose that Vi are themselves observations of independent multivariate normal random variables with mean 0 and known, nonsingular covariance matrix θ. Then α^ is admissible under squared error loss if r 2. Several estimators dominating α^ when r 3 are presented. Analogous results are presented for the case where σ2 or θ are unknown and some other generalizations are also considered. It is noted that some of these results for r3 appear in earlier papers of Baranchik and of Takada. {Vi} are ancillary statistics in the above setting. Hence admissibility of α^ depends on the distribution of the ancillary statistics, since if {Vi} is fixed instead of random, then α^ is admissible. This fact contradicts a widely held notion about ancillary statistics; some interpretations and consequences of this paradox are briefly discussed.

Comments

At the time of publication, author Lawrence Brown was affiliated with Cornell University. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania.

Keywords

admissibility, ancillary statistics, multiple linear regression

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Date Posted: 27 November 2017

This document has been peer reviewed.