Monday, January 29, 2001


Paul Eggermont
Department of Food and Resource Economics
University of Delaware

STRONG CONVEXITY AND NONPARAMETRIC FUNCTION ESTIMATION

The role of strongly convex minimization problems for nonparametric function estimation is surveyed. Under suitable circumstances maximum penalized likehood estimation problems are strongly convex problems in a reproducing kernel Hilbert space setting, or behave as if they were. The critical feature is that (ideally) it allows us to give error bounds for the (implicitly defined estimators in terms of explicitly defined kernel estimators. Some successful examples of this include maximum penalized likelihood density estimation using Good's first roughness penalization, least squares regression, such as spline smoothing (strongly convex), and total variation smoothing (where strong convexity can be faked), and ill-posed least squares problems with Tikhonov regularization. A reasonably unsuccessful case is the nonparametric deconvolution problem on the line, based on i.i.d\. data.