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Luc Devroye, Dominik Schäfer, László Györfi, Harro Walk

The estimation problem of minimum mean squared error

Regression analysis of a response variable Y requires careful selection of explanatory variables. The quality of a set of explanatory features X=(X⁽¹⁾, ..., X^(d)) can be measured in terms of the minimum mean squared error

L^*=min_fE{(Y−f(X))²}.

This paper investigates methods for estimating L^* from i.i.d. data. No estimate can converge rapidly for all distributions of (X,Y). For Lipschitz continuous regression function E{Y|X=x}, two estimators for L^* are discussed: fitting a regression estimate to a subset of the data and assessing its mean residual sum of squares on the remaining samples, and a nearest neighbor cross-validation type estimate.

Statistics & Decisions, Oldenbourg Wissenschaftsverlag

Print ISSN: 0721-2631
Volume: 21, 01/2003
Pages: 015

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