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Ideal for researchers and practitioners in statistics and industrial mathematics, this book covers the theory and practice of nonparametric estimation. It is novel in its use of maximum penalized likelihood estimation and convex minimization problem theory.
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 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.
From the reviews:
"This book is meant for specialized readers or graduate students interested in the theory, computation and application of Nonparametric Regression to real data, and the new contributions of the authors. ... For mathematically mature readers, the book would be a delight to read. ... The authors have not only written a scholarly and very readable book but provide major new methods and insights. ... it would help evaluate the methods as well as lead to teachable notes for a graduate course." (Jayanta K. Ghosh, International Statistical Review, Vol. 79 (1), 2011)
"This book is the second volume of a three-volume textbook in the Springer Series in Statistics. ... The second volume also belongs to the literature on nonparametric statistical inference and concentrates mainly on nonparametric regression. ... The book can be used for two main purposes: as a textbook for M.S./Ph.D. students in statistics, operations research, and applied mathematics, and as a tool for researchers and practitioners in these fields who want to develop and to apply nonparametric regression methods." (Yurij S. Kharin, Mathematical Reviews, Issue 2012 g)
"This book is meant for specialized readers or graduate students interested in the theory, computation and application of Nonparametric Regression to real data, and the new contributions of the authors. ... For mathematically mature readers, the book would be a delight to read. ... The authors have not only written a scholarly and very readable book but provide major new methods and insights. ... it would help evaluate the methods as well as lead to teachable notes for a graduate course." (Jayanta K. Ghosh, International Statistical Review, Vol. 79 (1), 2011)
"This book is the second volume of a three-volume textbook in the Springer Series in Statistics. ... The second volume also belongs to the literature on nonparametric statistical inference and concentrates mainly on nonparametric regression. ... The book can be used for two main purposes: as a textbook for M.S./Ph.D. students in statistics, operations research, and applied mathematics, and as a tool for researchers and practitioners in these fields who want to develop and to apply nonparametric regression methods." (Yurij S. Kharin, Mathematical Reviews, Issue 2012 g)