Regression models for high-dimensional data with correlated errors

Given a generalised linear regression model: y = Xβ + ε (1) where y is the n × 1 response vector; X is an n × p model matrix representing the predictors; and β is a p × 1 vector of coefficients to estimate. For mathematical simplicity, it is typical to set the first predictor as the intercept β0 so that the first column of X is the n×1 vector of ones. The intercept acts as a sink for the mean effect of included predictors, so one could remove the intercept term from the model by centering response and predictors. Unlike the classical conditions imposed on ε, we assume more generally that ε ∼ N(0,Σε) where Σε is a positive definite matrix. The variance-covariance matrix could be written in the form σ2εΩε so that we can obtain the classical model, σ2εI, as a convenient special case. The Generalised Least Squares (GLS) method focuses on the efficiency issue which fails in ordinary least squares. Efficient estimation of β in the generalised linear regression model requires knowledge of Ωε. To simplify the model, it is useful to consider cases in which Ωε is a known, symmetric, positive definite matrix. Since Ωε is a positive definite symmetric matrix, it can be decomposed into Ωε = CΛC where the parts of the decomposition C and Λ are the characteristic vectors (eigenvectors) and roots (eigenvalues) of Ωε, respectively. Now, let G ′ = CΛ−1/2, so Ω−1ε = G ′G where G is square and nonsingular. Premultiply the model (1) by G to obtain Gy = GXβ +Gε so y ∗ = X∗β + ε∗ The variance of ε ∗ is Var(ε∗) = Var(Gε) = σ2εGΩεG ′ = σ2εI so the Ordinary Least Squares (OLS) regression model can apply to this transformed model. Since Ωε is known, y ∗ and X∗n,p are observed data. In the classical model, ordinary least squares is efficient. Therefore, β ̂ = (X