An Ancillarity Paradox Which Appears in Multiple Linear Regression

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 α̂ 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 α̂ is admissible under squared error loss if r ≥ 2. Several estimators dominating α̂ 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 r ≥ 3 appear in earlier papers of Baranchik and of Takada. ᵢ\ are ancillary statistics in the above setting. Hence admissibility of α̂ depends on the distribution of the ancillary statistics, since if ᵢ\ is fixed instead of random, then α̂ is admissible. This fact contradicts a widely held notion about ancillary statistics; some interpretations and consequences of this paradox are briefly discussed.