Distribution-free properties of some asymptotic cumulants for the Mallows C p and its modifications in usual and ridge regression

Abstract

In multivariate multiple linear regression with a non-negative ridge parameter, when a model is underspecified, the asymptotic biases of the Mallows Cp and its modifications are derived up to order O(1) under non-normality. For a not underspecified model, the asymptotic biases are of smaller order and are shown to be distribution free. Similarly, under the latter condition, the common asymptotic variance of order O(1) for the statistics is distribution free. It is shown that the above results hold irrespective of the ridge parameter. Numerical illustrations with simulations under normality and non-normality give similar simulated results. These results justify the robust correct model selection under non-normality shown by simulations.