Estimation of a Mean Vector for Spherically Symmetric Distributions II: With a Residual

Abstract

In this chapter, we consider the canonical form of the general linear model introduced in Section 4.5 when a residual vector U is available. Recall that (X, U) is a random vector around (θ, 0) (such that dim X = dim θ = p and dim U = dim 0 = k) with a spherically symmetric distribution, that is, (X, U) ∼ SSp+k(θ, 0). Estimation of θ under quadratic loss ∥δ − θ∥2 parallels the normal situation presented in Sections 2.3 and 2.4 where \(X\sim {\mathcal N}_p(\theta , \sigma ^2 I_p)\) (with σ2 known) and the estimators of θ are of the form δ(X) = X + σ2g(X). In the case where σ2 is unknown see Section 2.4.3), the corresponding estimators are