Best Linear Unbiased Estimation in Linear Models

Introduction

In this article we consider the general linear model

$$\mathbf{y} = \mathbf{X}\beta + \epsilon,\quad \text{ or in short }\mathcal{M} =\{ \mathbf{y},\,\mathbf{X}\beta,\,{\sigma }^{2}\mathbf{V}\},$$

where X is a known n ×p model matrix, the vector y is an observable n-dimensional random vector, β is a p ×1 vector of unknown parameters, and ε is an unobservable vector of random errors with expectation E(ε) = 0, and covariance matrix cov(ε) = σ ² V, where σ ² > 0 is an unknown constant. The nonnegative definite (possibly singular) matrix V is known. In our considerations σ ² has no role and hence we may put σ ² = 1.

As regards the notation, we will use the symbols A′, A ⁻, A ⁺, $\mathcal{C}(\mathbf{A})$, $\mathcal{C}{(\mathbf{A})}^{\perp }$, and $\mathcal{N}(\mathbf{A})$ to denote, respectively, the transpose, a generalized inverse, the Moore–Penrose inverse, the column space, the orthogonal complement of the column space, and the null space, of the matrix A. By (A : B)...