A strongly consistent procedure for model selection in a regression problem

Abstract

We consider the multiple regression model Y_n= X_nβ+ E_n, where Y_n and E_n are n-vector random variables, Xn is an n×m matrix and β is an m-vector of unknown regression parameters. Each component of β may be zero or nonzero, which gives rise to 2^m possible models for multiple regression. We provide a decision rule for the choice of a model which is strongly consistent for the true model as n →∞. The result is proved under certain mild conditions, for instance without assuming normality of the distribution of the components of E_n.