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

Springer Texts in Statistics AJfred: Elements of Statistics for the Life and Social Sciences Berger: An Introduction to Probability and Stochastic Processes Bilodeau and Brenner: Theory of Multivariate Statistics BIom: 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 Fonnulae 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 Wassennan All of Statistics A Concise Course in Statistical Inference With 95 Figures , Springer

Larry W assennan Department of Statistics Carnegie Mellon University Baker Hali 228A 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 Cataloging-in-Publication Data Wasserman, Larry A. (Larry Alan), 1959- All of statistics: a concise course in statistica! inference 1 Larry a. W asserman. p. cm. - (Springer texts in statistics) Includes bibliographical references and index. 1. Mathematical statistics. 1. Title. QA276.12.W37 2003 519.5-dc21 Il. Series. 2003062209 ISBN 978-0-387-21736-9 (eBook) ISBN 978-1-4419-2322-6 DOI 10.1007/978-0-387-21736-9 © 2004 Springer Science+Business Media New York Originally published by Springer Science+Business Media, !ne in 2004 Softcover reprint of the hardcover 1 st edition 2004 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+ Business Media, LLC ), 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. 9 8 7 6 5 4 3 (Corrected second printing, 2005) springeronline.com

To Isa

Preface Taken literally, the title "All of Statistics" is an exaggeration. But in spirit, the title is apt, as the book does cover a much broader range of topics than a typical introductory book on mathematical statistics. This book is for people who want to learn probability and statistics quickly. It is suitable for graduate or advanced undergraduate students in computer science, mathematics, statistics, and related disciplines. The book includes modern topics like nonparametric curve estimation, bootstrapping, and clas- sification, topics that are usually relegated to follow-up courses. The reader is presumed to know calculus and a little linear algebra. No previous knowledge of probability and statistics is required. Statistics, data mining, and machine learning are all concerned with collecting and analyzing data. For some time, statistics research was con- ducted in statistics departments while data mining and machine learning re- search was conducted in computer science departments. Statisticians thought that computer scientists were reinventing the wheel. Computer scientists thought that statistical theory didn't apply to their problems. Things are changing. Statisticians now recognize that computer scientists are making novel contributions while computer scientists now recognize the generality of statistical theory and methodology. Clever data mining algo- rithms are more scalable than statisticians ever thought possible. Formal sta- tistical theory is more pervasive than computer scientists had realized. Students who analyze data, or who aspire to develop new methods for analyzing data, should be well grounded in basic probability and mathematical statistics. Using fancy tools like neural nets, boosting, and support vector

viii Preface machines without understanding basic statistics is like doing brain surgery before knowing how to use a band-aid. But where can students learn basic probability and statistics quickly? Nowhere. At least, that was my conclusion when my computer science colleagues kept asking me: "Where can I send my students to get a good understanding of modern statistics quickly?" The typical mathematical statistics course spends too much time on tedious and uninspiring topics (counting methods, two di- mensional integrals, etc.) at the expense of covering modern concepts (boot- strapping, curve estimation, graphical models, etc. ). So I set out to redesign our undergraduate honors course on probability and mathematical statistics. This book arose from that course. Here is a summary of the main features of this book. 1. The book is suitable for graduate students in computer science and honors undergraduates in math, statistics, and computer science. It is also useful for students beginning graduate work in statistics who need to fill in their background on mathematical statistics. 2. I cover advanced topics that are traditionally not taught in a first course. For example, nonparametric regression, bootstrapping, density estima- tion, and graphical models. 3. I have omitted topics in probability that do not play a central role in statistical inference. For example, counting methods are virtually ab- sent. 4. Whenever possible, I avoid tedious calculations in favor of emphasizing concepts. 5. I cover nonparametric inference before parametric inference. 6. I abandon the usual "First Term = Probability" and "Second Term = Statistics" approach . Some students only take the fi rst half and it would be a crime if they did not see any statistical theory. Furthermore, probability is more engaging when students can see it put to work in the context of statistics. An exception is the topic of stochastic processes which is included ill the later material. 7. T he course moves very quickly and covers much material. My colleagues joke that I cover all of statistics in this course and hence the title. The course is demanding but I have worked hard to make the material as intuitive as possible so that the material is very understandable despite the fast pace. 8. Rigor and clarity are not synonymous. I have tried to strike a good balance. To avoid getting bogged down in uninteresting technical details, many results are stated without proof. The bibliographic references at the end of each chapter point the student to appropriate sources.

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