On the degrees of freedom in shrinkage estimation

We study the degrees of freedom in shrinkage estimation of the regression coefficients. Generalizing the idea of the Lasso, we consider the problem of estimating the coefficients by the projection of the ordinary least squares estimator onto a closed convex set. Then an unbiased estimator of the degrees of freedom is derived in terms of geometric quantities under a smoothness condition on the boundary of the closed convex set. The result presented in this paper is applicable to estimation with a wide class of constraints. As an application, we obtain a Cp-type criterion and AIC for selecting the tuning parameter.