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A new class of biased estimates in linear regression. (English) Zbl 0784.62065
Summary: Consider the linear regression model \(Y=X \beta+ \varepsilon\), \(E \varepsilon=0\) and \(\text{Cov} (\varepsilon)=\sigma^ 2 I\). Motivated by an interpretation of ridge estimates \(\hat\beta_ R=(X' X+kI)^{-1}X' Y\), we propose a new class of biased estimates \(\hat\beta_ d=(X' X+I)^{-1}(X' Y+d \hat\beta)\) to combat multicollinearity, where \(0<d<1\) is a parameter and \(\hat\beta\) is the least squares estimate. \(\hat\beta_ d\) combines the advantages of \(\hat\beta_ R\) and Stein estimate \(\hat\beta_ s=c\hat\beta\).
Theory and simulation results show that \(\hat\beta_ d\) has similarly good properties as \(\hat\beta_ R\). The advantage of \(\hat\beta_ d\) over \(\hat\beta_ R\) is that \(\hat\beta_ d\) is a linear function of \(d\). So the selection of \(d\) is simple.

MSC:
62J07 Ridge regression; shrinkage estimators (Lasso)
62J05 Linear regression; mixed models
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