High-Dimensionality-Adjusted Asymptotically Loss- and Mean-Efficient GCp Criterion for Normal Multivariate Linear Regression Models

A variable selection method is put forward for multivariate linear regression models which obey normality. This method hinges on minimizing a generalized Cp (GCp) criterion which is defined by adding a positive constant value (the product of α and the number of parameters in the mean structure) to the minimum value of the multivariate residual sum of squares. The paper seeks to clarify the sufficient condition for α to simultaneously satisfy asymptotically loss- and mean-efficient properties in an asymptotic framework such that the sample size always goes to ∞, but the dimension of the vector of response variables can be either fixed or infinite. Based on this, we propose an asymptotically loss- and mean-efficient GCp criterion by using α, which satisfies the obtained sufficient condition even with high dimensionality of the vector of response variables.