Table Of Content

springer Series in Statistics Advisors: P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg, I. Olkin, N. Wermuth, S. Zeger Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo Springer Series in Statistics Andersen/Borgan/Gill/Keiding: Statistical Models Based on Counting Processes. Andrews/Herzberg: Data: A Collection of Problems from Many Fields for the Student and Research Worker. Anscombe: Computing in Statistical Science through APL. Berger: Statistical Decision Theory and Bayesian Analysis, 2nd edition. Bolfarine/Zacks: Prediction Theory for Finite Populations. Borg/Groenen: Modem Multidimensional Scaling: Theory and Applications Bremaud: Point Processes and Queues: Martingale Dynamics. BrockwelUDavis: Time Series: Theory and Methods, 2nd edition. Daley/Vere-Jones: An Introduction to the Theory of Point Processes. Dzhaparidze: Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series. Efromovich: Nonparametric Curve Estimation: Methods, Theory, and Applications. Fahrmeir/Tutz: Multivariate Statistical Modelling Based on Generalized Linear Models. Farebrother: Fitting Linear Relationships: A History of the Calculus of Observations 1750-1900. Farrell: Multivariate Calculation. Federer: Statistical Design and Analysis for Intercropping Experiments, Volume I: Two Crops. Federer: Statistical Design and Analysis for Intercropping Experiments, Volume II: Three or More Crops. Fienberg/Hoaglin/Kruskal/Tanur(Eds.): A Statistical Model: Frederick Mosteller's Contributions to Statistics, Science and Public Policy. Fisher/Sen: The Collected Works of Wassily Hoeffding. Good: Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses. Goodman/Kruskal: Measures of Association for Cross Classifications. Gourieroux: ARCH Models and Financial Applications. Grandell: Aspects of Risk Theory. Haberman: Advanced Statistics, Volume I: Description of Populations. Hall: The Bootstrap and Edgeworth Expansion. Hdrdle: Smoothing Techniques: With Implementation in S. Hart: Nonparametric Smoothing and Lack-of-Fit Tests. Hartigan: Bayes Theory. Hedayat/Sloane/Stufken: Orthogonal Arrays: Theory and Applications. Heyde: Quasi-Likelihood and its Application: A General Approach to Optimal Parameter Estimation. Heyer: Theory of Statistical Experiments. Huet/Bouvier/Gruet/Jolivet: Statistical Tools for Nonlinear Regression: A Practical Guide with S-PLUS Examples. Jolliffe: Principal Component Analysis. Kolen/Brennan: Test Equating: Methods and Practices. Kotz/Johnson (Eds.): Breakthroughs in Statistics Volume I. (continued after index) Springer Series in Statistics (continued from p. ii) Kotz/Johnson (Eds.): Breakthroughs in Statistics Volume II. Kotz/Johnson (Eds.): Breakthroughs in Statistics Volume III. Kres: Statistical Tables for Multivariate Analysis. Kiichler/S0rensen: Exponential Families of Stochastic Processes. Le Cam: Asymptotic Methods in Statistical Decision Theory. Le Cam/Yang: Asymptotics in Statistics: Some Basic Concepts. Longford: Models for Uncertainty in Educational Testing. Manoukian: Modern Concepts and Theorems of Mathematical Statistics. Miller, Jr.: Simultaneous Statistical Inference, 2nd edition. MostellerAVallace: Applied Bayesian and Classical Inference: The Case of the Federalist Papers. Parzen/Tanabe/Kitagawa: Selected Papers of Hirotugu Akaike. Politis/Romano/Wolf: Subsampling. Pollard: Convergence of Stochastic Processes. Pratt/Gibbons: Concepts of Nonparametric TTieory. Ramsay/Silverman: Functional Data Analysis. Rao/Toutenburg: Linear Models: Least Squares and Alternatives. Read/Cressie: Goodness-of-Fit Statistics for Discrete Multivariate Data. Reinsel: Elements of Multivariate Time Series Analysis, 2nd edition. Reiss: A Course on Point Processes. Reiss: Approximate Distributions of Order Statistics: With Applications to Non-parametric Statistics. Rieder: Robust Asymptotic Statistics. Rosenbaum: Observational Studies. Ross: Nonlinear Estimation. Sachs: Applied Statistics: A Handbook of Techniques, 2nd edition. Sdmdal/Swensson/Wretman: Model Assisted Survey Sampling. Schervish: Theory of Statistics. Seneta: Non-Negative Matrices and Markov Chains, 2nd edition. Shao/Tu: The Jackknife and Bootstrap. Siegmund: Sequential Analysis: Tests and Confidence Intervals. Simonojf: Smoothing Methods in Statistics. Singpurwalla and Wilson: Statistical Methods in Software Engineering: Reliability and Risk. Small: The Statistical Theory of Shape. Stein: Interpolation of Spatial Data: Some Theory for Kriging Tanner: Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, 3rd edition. Tong: The Multivariate Normal Distribution. van der Vaart/Wellner: Weak Convergence and Empirical Processes: With Applications to Statistics. Vapnik: Estimation of Dependences Based on Empirical Data. Weerahandi: Exact Statistical Methods for Data Analysis. West/Harrison: Bayesian Forecasting and Dynamic Models, 2nd edition. Walter: Introduction to Variance Estimation. Yaglom: Correlation Theory of Stationary and Related Random Functions I: Basic Results. Sam Efromovich Nonparametric Curve Estimation Methods, Theory, and AppUcations With 130 Figures Springer Sam Efromovich Department of Mathematics and Statistics University of New Mexico Albuquerque, NM 87131-1141 USA Library of Congress Cataloging-in-Publication Data Efromovich, Sam. Nonparametric curve estimation : methods, theory, and applications / Sam Efromovich. p. cm. — (Springer series in statistics) Includes bibliographical references and index. ISBN 0-387-98740-1 (hardcover) 1. Nonparametric statistics. 2. Estimation theory. 1. Title. II. Series. QA278.8.E35 1999 519.5—dc21 99-13253 © 1999 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, 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, computer software, or by similar or dissimilar methodology now known or here- after developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-387-98740-1 Springer-Verlag New York Berlin Heidelberg SPIN 10709119 To my parents Preface Appropriate for a one-semester course, this self-contained book is an introduction to nonparametric curve estimation theory. It may be used for teachinggraduatestudentsinstatistics(inthiscaseanintermediatecourse in statistical inference, on the level of the book by Casella and Berger (1990), is the prerequisite) as well as for diverse classes with students from other sciences including engineering, business, social, medical, and biolog- ical among others (in this case a traditional intermediate calculus course plusanintroductorycourseinprobability,onthelevelofthebookbyRoss (1997), are the prerequisites). There are several distinguishing features of this book that should be highlighted: -Allbasicstatisticalmodels,includingprobabilitydensityestimation,nonparametric regression, time series analysis including spectral analysis, and filtering of time-continuous signals, are considered as one general problem. As a result, universal methods of estimation are discussed, and students become familiar with a wide spectrum of applications of nonparametric methods. - Main emphasis is placed on the case of small sample sizes and data- driven orthogonal series estimates (Chapters 1–6). Chapter 7 discusses (with proofs) modern asymptotic results, and Chapter 8 is devoted to a thorough discussion of nonseries methods. - The companion software package (available over the World Wide Web) allowsstudentstoproduceandmodifyalmostallfiguresofthebookaswell as to analyze a broad spectrum of simulated and real data sets. Based on the S–PLUS environment, this package requires no knowledge of S–PLUS viii Preface and is elementary to use. Appendix B explains how to install and use this package;italsocontainsinformationabouttheaffordableS–PLUSStudent Edition for PC. -“PracticalSeminar”sectionsaredevotedtoapplyingthemethodsstudied to the analysis and presentation of real data sets. The software for these sections allows students to analyze any data set that exists in the S–PLUS environment. - “Case Study” sections allow students to explore applications of basic methods to more complicated practical problems. These sections together with “Special Topic” sections give the instructor some flexibility in choos- ing additional material beyond the core. - Plenty of exercises with different levels of difficulty will allow the instructor to keep students with different mathematical and statistical backgrounds out of trouble! -“Notes”sectionsattheendofeachchapterareprimarilydevotedtobooks for further reading. They also capture some bibliographic comments, side issues, etc. - Appendix A contains a brief review of fundamentals of statistical inference. All the related notions and notations used in the book may be found there. It is highly recommended to review these fundamentals prior to studying Chapters 3–8. Also, exercises for Appendix A may be used as a first quiz or homework. Abitofadvicetothereaderwhowouldliketousethisbookforself-study and who is venturing for the first time into this area. You can definitely just read this book as any other text without using the companion software. There are plenty of figures (more than a hundred), which will guide you through the text. However, if you have decided to study nonparamet- rics, then you are probably interested in data analysis. I cannot stress too strongly the importance of combining reading with analyzing both simulated and real data sets. This is the kind of experience that you can gain only via repeated exercises, and here the software can make this process dramatically quicker and less painful. Using the software will allow you to check virtually every claim and development mentioned in the book and make the material fully transparent. Also, please review the fundamentals outlined in Appendix A prior to studying Chapters 3–8. All further developments related to this book will be posted on the WWW page http://www.math.unm.edu/∼efrom/book1, and the author may be contacted by electronic mail as [email protected]. Acknowledgments I would like to thank everyone who in various ways has had influence on thisbook.MybiggestthanksgotoMarkPinsker.AlexSamarovgraciously read and gave comments on a draft of the book. John Kimmel provided Preface ix invaluable assistance through the publishing process. Three “generations” ofmystudentswhotooktheclassonnonparametriccurveestimationbased on this book shared with me their thoughts, comments, and suggestions. I thank all of you. Sam Efromovich Albuquerque, USA, 1999