Best Linear Unbiased Estimation in Linear Models

where X is a known n × p model matrix, the vector y is an observable ndimensional 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) = σV, where σ > 0 is an unknown constant. The nonnegative definite (possibly singular) matrix V is known. In our considerations σ has no role and hence we may put σ = 1. As regards the notation, we will use the symbolsA,A,A, C (A), C (A), and N (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) we denote the partitioned matrix with A and B as submatrices. By A we denote any matrix satisfying C (A) = N (A) = C (A). Furthermore, we will write PA = AA = A(AA)A to denote the orthogonal projector (with respect to the standard inner product) onto C (A). In particular, we denote H = PX and M = In −H. One choice for X ⊥ is of course the projector M.