Degrees of Freedom for Piecewise Lipschitz Estimators

A representation of the degrees of freedom akin to Stein's lemma is given for a class of estimators of a mean value parameter in $\mathbb{R}^n$. Contrary to previous results our representation holds for a range of discontinues estimators. It shows that even though the discontinuities form a Lebesgue null set, they cannot be ignored when computing degrees of freedom. Estimators with discontinuities arise naturally in regression if data driven variable selection is used. Two such examples, namely best subset selection and lasso-OLS, are considered in detail in this paper. For lasso-OLS the general representation leads to an estimate of the degrees of freedom based on the lasso solution path, which in turn can be used for estimating the risk of lasso-OLS. A similar estimate is proposed for best subset selection. The usefulness of the risk estimates for selecting the number of variables is demonstrated via simulations with a particular focus on lasso-OLS.Comment: 113 pages, 89 figure

Full-field hard x-ray microscopy with interdigitated silicon lenses

Full-field x-ray microscopy using x-ray objectives has become a mainstay of the biological and materials sciences. However, the inefficiency of existing objectives at x-ray energies above 15 keV has limited the technique to weakly absorbing or two-dimensional (2D) samples. Here, we show that significant gains in numerical aperture and spatial resolution may be possible at hard x-ray energies by using silicon-based optics comprising 'interdigitated' refractive silicon lenslets that alternate their focus between the horizontal and vertical directions. By capitalizing on the nano-manufacturing processes available to silicon, we show that it is possible to overcome the inherent inefficiencies of silicon-based optics and interdigitated geometries. As a proof-of-concept of Si-based interdigitated objectives, we demonstrate a prototype interdigitated lens with a resolution of ~255 nm at 17 keV.Comment: 10 pages, 5 figure. Submitted to Applied Physics Letters 31st March 2015, rejected 17th June 201