This is the second volume of a text on the theory and practice of maximum penalized likelihood estimation. It is intended for graduate students in s- tistics, operationsresearch, andappliedmathematics, aswellasresearchers and practitioners in the ?eld. The present volume was supposed to have a short chapter on nonparametric regression but was intended to deal mainly with inverse problems. However, the chapter on nonparametric regression kept growing to the point where it is now the only topic covered. Perhaps there will be a Volume III. It might even deal with inverse problems. But for now we are happy to have ?nished Volume II. The emphasis in this volume is on smoothing splines of arbitrary order, but other estimators (kernels, local and global polynomials) pass review as well. We study smoothing splines and local polynomials in the context of reproducing kernel Hilbert spaces. The connection between smoothing splines and reproducing kernels is of course well-known. The new twist is thatlettingtheinnerproductdependonthesmoothingparameteropensup new possibilities: It leads to asymptotically equivalent reproducing kernel estimators (without quali?cations) and thence, via uniform error bounds for kernel estimators, to uniform error bounds for smoothing splines and, via strong approximations, to con?dence bands for the unknown regression function. ItcameassomewhatofasurprisethatreproducingkernelHilbert space ideas also proved useful in the study of local polynomial estimators.

This book is intended for graduate students in statistics and industrial mathematics, as well as researchers and practitioners in the field. It covers both theory and practice of nonparametric estimation. The text is novel in its use of maximum penalized likelihood estimation, and the theory of convex minimization problems (fully developed in the text) to obtain convergence rates. A substantial effort has been made to discuss computational details, and to include simulation studies and analyses of some classical data sets using fully automatic (data driven) procedures.

Nonparametric Regression.- Smoothing Splines.- Kernel Estimators.- Sieves.- Local Polynomial Estimators.- Other Nonparametric Regression Problems.- Smoothing Parameter Selection.- Computing Nonparametric Estimators.- Kalman Filtering for Spline Smoothing.- Equivalent Kernels for Smoothing Splines.- Strong Approximation and Confidence Bands.- Nonparametric Regression in Action.