Selection of the number of regression variables; A minimax choice of generalized FPE
Annals of the Institute of Statistical Mathematics, 38 (1):
459--474 (December 1986)DOI: 10.1007/BF02482533
Abstract
SummaryA generalized Final Prediction Error (FPEα)\_ criterion is considered. Based onn observations, the numberk of regression variables is selected from a given range 0≦k≦K, so as to minimizeFPEα(k)=nσ̂ 2(k)+αk\nσ̂ 2(k)/(n−K)\FPEα(k)=nσˆ2(k)+αk\nσˆ2(k)/(n−K)\FPE\_\textbackslashalpha (k) = n\textbackslashhat \textbackslashsigma ˆ2 (k) + \textbackslashalpha k\textbackslash\ n\textbackslashhat \textbackslashsigma ˆ2 (k)/(n - K)\textbackslash\ . It is shown that if α tends to infinity withn, the selection is consistent but the maximum of the mean squared error of estimates of parameters diverges to infinity with the same order of divergence as that of α. A meaningful minimax choice of α exists for a regret type mean squared error, while for simple mean squared error it is trivially 0. The minimax regret choice of α converges to a constant, approximately 3.5 forK≧8 ifn−K increases simultaneously withn, otherwise it diverges to infinity withn.