Selection of the number of regression variables; A minimax choice of generalized FPE
type: Item
instance of: scholarly article
title: Selection of the number of regression variables; A minimax choice of generalized FPE
zbMATH Open document ID: 0619.62060
DOI: 10.1007/BF02482533
author: Ritei Shibata
published in: Annals of the Institute of Statistical Mathematics
publication date: 1986-01-01T00:00:00Z
review text: A generalized Final Prediction Error \((FPE_{\alpha})\) criterion is considered. Based on n observations, the number k of regression variables is selected from a given range \(0\leq k\leq K\), so as to minimize \(FPE_{\alpha}(k)=n{\hat \sigma}^ 2(k)+\alpha k\{n{\hat \sigma}^ 2(k)/(n-K)\}\). It is shown that if \(\alpha\) tends to infinity with n, 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 \(\alpha\). A meaningful minimax choice of \(\alpha\) exists for a regret type mean squared error, while for simple mean squared error it is trivially 0. The minimax regret choice of \(\alpha\) converges to a constant, approximately 3.5 for \(K\geq 8\) if n-K increases simultaneously with n, otherwise it diverges to infinity with n.
Mathematics Subject Classification ID: 62J05
zbMATH DE Number: 4005372
zbMATH Keywords: selection of the number of regression variables
zbMATH Keywords: generalized FPE
zbMATH Keywords: nested models
zbMATH Keywords: AIC
zbMATH Keywords: generalized Final Prediction Error
zbMATH Keywords: mean squared error of estimates
zbMATH Keywords: regret type mean squared error
zbMATH Keywords: minimax regret choice
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02482533
OpenAlex ID: W2089299406
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description: scientific article; zbMATH DE number 4005372