Abstract

The generalized and smooth James-Stein thresholding functions link and extend the thresholding functions employed by the James-Stein estimator, the block- and adaptive-lasso in variable selection, and the soft-, hard- and block-thresholding in wavelet smoothing. The estimator is indexed by two hyperparameters for more flexibility and a smoothness parameter for better estimation of its ℓ2-risk with the Stein unbiased risk estimate (SURE). For blocks of a fixed size, a situation that arises when observing concomitant signals (e.g., gravitational wave bursts), we derive a universal threshold, an information criterion and an oracle inequality for block thresholding. Smooth James-Stein thresholding can also be employed in parametric regression for variable selection. In that case a unique smooth estimate is defined, its smooth SURE is derived, which provides the equivalent degrees of freedom of adaptive lasso as a side result. The new estimator enjoys smoothness like ridge regression and performs variable selection like lasso.