Abstract: We study the asymptotic optimality of C_P-type criteria from the perspective of prediction in high-dimensional multivariate linear regression models, where the dimension of a response matrix is large, but does not exceed the sample size. We derive conditions in order that the generalized C_P (GC_P) exhibits asymptotic loss efficiency (ALE) and asymptotic mean efficiency (AME) in such high-dimensional data. Moreover, we clarify that one of the conditions is necessary for GC_P to exhibit both ALE and AME. As a result, we show that the modified C_P can claim both ALE and AME, but the original C_P cannot in high-dimensional data. The finite-sample performance of the GC_P with several tuning parameters is compared by means of a simulation study.

Key words and phrases: Asymptotic theory, high-dimensional statistical inference, model selection/variable selection.