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Springer Texts in Statistics Advisors: George Casella Stephen Fienberg Ingram Olkin

Springer Texts in Statistics Alfred: Elements of Statistics for the Life and Social Sciences Berger: An Introduction to Probability and Stochastic Processes Bilodeau and Brenner: Theory of Multivariate Statistics Blom: Probability and Statistics: Theory and Applications Brockwell and Davis: Introduction to Times Series and Forecasting, Second Edition Chow and Teicher: Probability Theory: Independence, Interchangeability, Martingales, Third Edition Christensen: Advanced Linear Modeling: Multivariate, Time Series, and Spatial Data—Nonparametric Regression and Response Surface Maximization, Second Edition Christensen: Log-Linear Models and Logistic Regression, Second Edition Christensen: Plane Answers to Complex Questions: The Theory of Linear Models, Third Edition Creighton: A First Course in Probability Models and Statistical Inference Davis: Statistical Methods for the Analysis of Repeated Measurements Dean and Voss: Design and Analysis of Experiments du Toit, Steyn, and Stumpf: Graphical Exploratory Data Analysis Durrett: Essentials of Stochastic Processes Edwards: Introduction to Graphical Modelling, Second Edition Finkelstein and Levin: Statistics for Lawyers Flury: A First Course in Multivariate Statistics Jobson: Applied Multivariate Data Analysis, Volume I: Regression and Experimental Design Jobson: Applied Multivariate Data Analysis, Volume II: Categorical and Multivariate Methods Kalbfleisch: Probability and Statistical Inference, Volume I: Probability, Second Edition Kalbfleisch: Probability and Statistical Inference, Volume II: Statistical Inference, Second Edition Karr: Probability Keyfitz: Applied Mathematical Demography, Second Edition Kiefer: Introduction to Statistical Inference Kokoska and Nevison: Statistical Tables and Formulae Kulkarni: Modeling, Analysis, Design, and Control of Stochastic Systems Lange: Applied Probability Lehmann: Elements of Large-Sample Theory Lehmann: Testing Statistical Hypotheses, Second Edition Lehmann and Casella: Theory of Point Estimation, Second Edition Lindman: Analysis of Variance in Experimental Design Lindsey: Applying Generalized Linear Models (continued after index)

Larry Wasserman All of Nonparametric Statistics With 52 Illustrations

Larry Wasserman Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213-3890 USA larry@stat.cmu.edu Editorial Board George Casella Department of Statistics University of Florida Gainesville, FL 32611-8545 USA Stephen Fienberg Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213-3890 USA Ingram Olkin Department of Statistics Stanford University Stanford, CA 94305 USA Library of Congress Control Number: 2005925603 ISBN-10: 0-387-25145-6 ISBN-13: 978-0387-25145-5 Printed on acid-free paper. © 2006 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, com- puter software, or by similar or dissimilar methodology now known or hereafter developed is for- bidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (MVY) 9 8 7 6 5 4 3 2 1 springeronline.com

To Isa

Preface There are many books on various aspects of nonparametric inference such as density estimation, nonparametric regression, bootstrapping, and wavelets methods. But it is hard to ﬁnd all these topics covered in one place. The goal of this text is to provide readers with a single book where they can ﬁnd a brief account of many of the modern topics in nonparametric inference. The book is aimed at master’s-level or Ph.D.-level statistics and computer science students. It is also suitable for researchers in statistics, machine learn- ing and data mining who want to get up to speed quickly on modern non- parametric methods. My goal is to quickly acquaint the reader with the basic concepts in many areas rather than tackling any one topic in great detail. In the interest of covering a wide range of topics, while keeping the book short, I have opted to omit most proofs. Bibliographic remarks point the reader to references that contain further details. Of course, I have had to choose topics to include and to omit, the title notwithstanding. For the most part, I decided to omit topics that are too big to cover in one chapter. For example, I do not cover classiﬁcation or nonparametric Bayesian inference. The book developed from my lecture notes for a half-semester (20 hours) course populated mainly by master’s-level students. For Ph.D.-level students, the instructor may want to cover some of the material in more depth and require the students to ﬁll in proofs of some of the theorems. Throughout, I have attempted to follow one basic principle: never give an estimator without giving a conﬁdence set.

viii Preface The book has a mixture of methods and theory. The material is meant to complement more method-oriented texts such as Hastie et al. (2001) and Ruppert et al. (2003). After the Introduction in Chapter 1, Chapters 2 and 3 cover topics related to the empirical cdf such as the nonparametric delta method and the bootstrap. Chapters 4 to 6 cover basic smoothing methods. Chapters 7 to 9 have a higher theoretical content and are more demanding. The theory in Chapter 7 lays the foundation for the orthogonal function methods in Chapters 8 and 9. Chapter 10 surveys some of the omitted topics. I assume that the reader has had a course in mathematical statistics such as Casella and Berger (2002) or Wasserman (2004). In particular, I assume that the following concepts are familiar to the reader: distribution functions, convergence in probability, convergence in distribution, almost sure conver- gence, likelihood functions, maximum likelihood, conﬁdence intervals, the delta method, bias, mean squared error, and Bayes estimators. These back- ground concepts are reviewed brieﬂy in Chapter 1. Data sets and code can be found at: www.stat.cmu.edu/∼larry/all-of-nonpar I need to make some disclaimers. First, the topics in this book fall under the rubric of “modern nonparametrics.” The omission of traditional methods such as rank tests and so on is not intended to belittle their importance. Sec- ond, I make heavy use of large-sample methods. This is partly because I think that statistics is, largely, most successful and useful in large-sample situations, and partly because it is often easier to construct large-sample, nonparamet- ric methods. The reader should be aware that large-sample methods can, of course, go awry when used without appropriate caution. I would like to thank the following people for providing feedback and sugges- tions: Larry Brown, Ed George, John Laﬀerty, Feng Liang, Catherine Loader, Jiayang Sun, and Rob Tibshirani. Special thanks to some readers who pro- vided very detailed comments: Taeryon Choi, Nils Hjort, Woncheol Jang, Chris Jones, Javier Rojo, David Scott, and one anonymous reader. Thanks also go to my colleague Chris Genovese for lots of advice and for writing the LATEX macros for the layout of the book. I am indebted to John Kimmel, who has been supportive and helpful and did not rebel against the crazy title. Finally, thanks to my wife Isabella Verdinelli for suggestions that improved the book and for her love and support. Larry Wasserman Pittsburgh, Pennsylvania July 2005

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