Consider an experiment in which \( p \) characteristics of \( n \) samples are measured. The data from this experiment are denoted \( \mathbf{X} \), with \( \mathbf{X} \) as above. The matrix \( \mathbf{X} \) is called the design matrix. Additional information of the samples is available in the form of \( \mathbf{Y} \) (also as above). The variable \( \mathbf{Y} \) is generally referred to as the response variable. The aim of regression analysis is to explain \( \mathbf{Y} \) in terms of \( \mathbf{X} \) through a functional relationship like \( Y_i = f(\mathbf{X}_{i,\ast}) \). When no prior knowledge on the form of \( f(\cdot) \) is available, it is common to assume a linear relationship between \( \mathbf{X} \) and \( \mathbf{Y} \). This assumption gives rise to the linear regression model where \( \beta = (\beta_1, \ldots, \beta_p)^{\top} \) is the regression parameter. The parameter \( \beta_j \), \( j=1, \ldots, p \), represents the effect size of covariate \( j \) on the response. That is, for each unit change in covariate \( j \) (while keeping the other covariates fixed) the observed change in the response is equal to \( \beta_j \).