Rejoinder: An Ancillarity Paradox which Appears in Multiple Linear Regression

Abstract

Consider a multiple linear regression in which Yi, i = 1, ?, n, are independent normal variables with variance s2 and E(Yi) = a + V'i�, where Vi ? Rr and � ? Rr. Let a^ denote the usual least squares estimator of a. Suppose that Vi are themselves observations of independent multivariate normal random variables with mean 0 and known, nonsingular covariance matrix ?. Then a^ is admissible under squared error loss if r = 2. Several estimators dominating a^ when r = 3 are presented. Analogous results are presented for the case where s2 or ? are unknown and some other generalizations are also considered. It is noted that some of these results for r = 3 appear in earlier papers of Baranchik and of Takada. {Vi} are ancillary statistics in the above setting. Hence admissibility of a^ depends on the distribution of the ancillary statistics, since if {Vi} is fixed instead of random, then a^ is admissible. This fact contradicts a widely held notion about ancillary statistics; some interpretations and consequences of this paradox are briefly discussed.