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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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