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Mathematical Statistics, the Authors & Onker Describe Key Concepts from Statistics and Give a Mathematical Basis for Important Statistical Methods B Statistics is the science that focuses on drawing conclusions from ijma data, by modeling and analyzing the data using probabilistic , J models. In An Introduction to Mathematical Statistics, the authors onker describe key concepts from statistics and give a mathematical basis for important statistical methods. Much attention is paid to & the sound application of those methods to data. V an An Introduction to The three main topics in statistics are estimators, tests, and der confidence regions. The authors illustrate these in many examples, V aart with a separate chapter on regression models, including linear Mathematical regression and analysis of variance. They also discuss the optimality of estimators and tests, as well as the selection of the best-fitting Statistics Mathematical to An Introduction model. Each chapter ends with a case study in which the described statistical methods are applied. Statistics This book assumes a basic knowledge of probability theory, calculus, and linear algebra. Fetsje Bijma worked as an assistant professor of mathematics at the Vrije Universiteit Amsterdam for 10 years. She studied mathematics at the University of Groningen and received a Ph.D. at the Faculty of Medicine of the Vrije Universiteit Amsterdam. She now works in industry. Fetsje Bijma Marianne Jonker is a biostatistician at the Radboud University Medical Center in Nijmegen. She studied mathematics and received her Ph.D. at the Marianne Jonker Vrije Universiteit Amsterdam, where she worked as an assistant professor Aad van der Vaart in mathematics for some time. She has done work as a biostatistician for the Leiden University Medical Center in Leiden and the VU University Medical Center in Amsterdam. Aad van der Vaart is a professor of stochastics at Leiden University and member of the Royal Netherlands Academy of Arts and Sciences. He studied mathematics and psychology at Leiden University and received a Ph.D. in mathematics at the same university. He held positions at universities in the United States and in Paris (France) and at the Vrije Universiteit Amsterdam. 9789462985100 Bijma - Mathematical Statistics - def - 02-05-2017.indd 1 2-5-2017 15:30:03 An Introduction to Mathematical Statistics An Introduction to Mathematical Statistics Fetsje Bijma, Marianne Jonker, Aad van der Vaart Amsterdam University Press Original publication: Fetsje Bijma, Marianne Jonker, Aad van der Vaart, Inleiding in de statistiek. Epsilon Uitgaven, 2016 [ISBN 978-90-5041-135-6] © Fetsje Bijma, Marianne Jonker, Aad van der Vaart, 2016 Translated by: Reinie Erné Cover design: V3-Services, Baarn Amsterdam University Press English-language titles are distributed in the US and Canada by the University of Chicago Press. ISBN 978 94 6298 510 0 e-ISBN 978 90 4853 611 5 (pdf) NUR 916 DOI 10.5117/9789462985100 © Fetsje Bijma, Marianne Jonker, Aad van der Vaart / Amsterdam University Press B.V., Amsterdam 2017 All rights reserved. Without limiting the rights under copyright reserved above, no part of this book may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without the written permission of both the copyright owner and the author of the book. Every effort has been made to obtain permission to use all copyrighted illustrations reproduced in this book. Nonetheless, whosoever believes to have rights to this material is advised to contact the publisher. PREFACE This book gives an introduction into mathematical statistics. It was written for bachelor students in (business) mathematics, econometrics, or any other subject with a solid mathematical component. We assume that the student already has solid knowledge of probability theory to the extent of a semester course at the same level. In Chapter 1, we give the definition and several examples of a statistical model, the foundation of every statistical procedure. Some techniques from descriptive statistics that can assist in setting up and validating statistical models are discussed in Chapter 2. The following chapters discuss the three main topics in mathematical statistics: estimating, testing, and constructing confidence regions. These subjects are discussed in Chapters 3, 4, and 5, respectively. Next, Chapter 6 provides deeper theoretical insight, in particular into the question under what circumstances and in what sense certain statistical models are mathematically optimal. In Chapter 7, we describe several regression models that are commonly used in practice. The theory from the previous chapters is applied to estimate and test unknown model parameters and give confidence regions for them. Finally, in Chapter 8, we discuss model selection. In that chapter, various criteria are presented that can be used to find the best-fitting model from a collection of (regression) models. Sections and examples markedwitha*aremoredifficult and do not belong to the basic subject matter of mathematical statistics. Every chapter concludes with a summary. In Appendix A, we recall elements from probability theory that are relevant for understanding the subject matter of this book. In Appendix B, we discuss properties of the multivariate normal distribution, which is used in several sections. Appendix C contains tables with values of distribution and quantile functions of several distributions to which we refer in the text. These are meant to be used at home or during problem sessions. In “real life,” these tables are no longer used: the computer is faster, more accurate, and easier to use. The statistical package R, for example, contains standard functions for the distribution function, the density function, and the quantile function of all standard distributions. The mathematical style of this book is more informal than that of many mathematics books. Theorems and lemmas are not always proved or may be formulated in an informal manner. The reason is that a pure mathematical treatment is only possible using measure theory, of which we do not assume any knowledge. On the other hand, the relevance and motivation of the theorems are also clear without going into all the details. Each chapter concludes with a case study. It often contains a statistical problem that is answered as well as possible based on the collected data, using the statistical techniques and methods available at that point in the book. The R-code and data of these applications, as well as the data of several case studies described in the book, are available and can be downloaded from the book’s webpage at http://www.aup.nl. Though this book includes examples, practice is indispensable to gain insight into the subject matter. The exercises at the end of each chapter include both theoretical and more practically oriented problems. Appendix D contains short answers to most v exercises. Solutions that consist of a proof are not included. FURTHER READING The book has taken form over a period of 20 years. It was originally written in Dutch and used yearly for the course “Algemene Statistiek” (General Statistics) for (business) mathematics and econometrics students given by the mathematics Reference [1] is an introduction to many aspects of statistics, somewhat comparable department of VU University Amsterdam. The various lecturers of the course to An Introduction to Mathematical Statistics. References [3] and [4] are standard contributed to the book to a greater or lesser extent. One of them is Bas Kleijn. We books that focus more on mathematical theory, and estimation and tests, respec- want to thank him for his contribution to the appendix on probability theory. More than tively. Reference [6] describes the use of asymptotic methods in statistics, on a 2000 students have studied the book. Their questions on the subject and advice on the higher mathematical level, and gives several proofs left out in An Introduction to presentation have helped give the book its present form. They have our thanks. The Mathematical Statistics. Reference [5] is a good starting point for whoever wants to starting point of the book was the syllabus “Algemene Statistiek” (General Statistics) delve further into the Bayesian thought process, and reference [7] provides the same of J. Oosterhoff, professor of mathematical statistics at VU University Amsterdam for nonparametric methods, which are mentioned in An Introduction to Mathematical until the mid-’90s. We dedicate this book to him. Statistics but perhaps less prominently than in current practice. Reference [2] In 2013, the first edition of this book was published in Dutch, and three elaborates on the relevance of modeling using regression models, for example to draw years later, in 2016, the second Dutch edition came out. This second edition has causal conclusions in economic or social sciences. been translated into English, with some minor changes. We thank Reinie Ern´efor translation. [1] Davison, A.C., (2003). Statistical models. Cambridge University Press. Amsterdam and Leiden, March 2017 [2] Freedman, D., (2005). Statistical models: theory and applications. Cambridge University Press. [3] Lehmann, E.L. and Casella, G., (1998). Theory of point estimation. Springer. [4] Lehmann, E.L. and Romano, J.P., (2005). Testing statistical hypotheses. Springer. [5] Robert, C.P., (2001). The Bayesian choice. Springer-Verlag. [6] van der Vaart, A.W., (1998). Asymptotic statistics. Cambridge University Press. [7] Wasserman, L., (2005). All of nonparametric statistics. Springer. vi vii exercises. Solutions that consist of a proof
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