The Skew-Normal and Related Families Interest in the skew-normal and related families of distributions has grown enormously over recent years, as theory has advanced, challenges of data have grown and computational tools have become more readily available. This comprehensive treatment, blending theory and practice, will be the standard resource for statisticians and applied researchers. Assuming only basic knowledge of (non-measure-theoretic) probability and statistical inference, the book is accessible to the wide range of researchers who use statistical modelling techniques. Guiding readers through the main concepts and results, the book covers both the probability and the statistics sides of the subject, in the univariate and multivariate settings. The theoretical development is complemented by numerous illustrations and applications to a range of fields including quantitative finance, medical statistics, environmental risk studies and industrial and business efficiency. The authors’ freely available R package sn, available from CRAN, equips readers to put the methods into action with their own data. adelchi azzalini was Professor of Statistics in the Department of Statistical Sciences at the University of Padua until his retirement in 2013. Over the last 15 years or so, much of his work has been dedicated to the research area of this book. He is regarded as the pioneer of this subject due to his 1985 paper on the skew-normal distribution; in addition, several of his subsequent papers, some of which have been written jointly with Antonella Capitanio, are considered to represent fundamental steps. He is the author or co-author of three books, over 70 research papers and four packages written in the R language. antonella capitanio is Associate Professor of Statistics in the Department of Statistical Sciences at the University of Bologna. She began working on the skew-normal distribution about 15 years ago, co-authoring with Adelchi Azzalini a series of papers, related to the skew-normal and skew-elliptical distributions, which have provided key results in this area. INSTITUTE OF MATHEMATICAL STATISTICS MONOGRAPHS Editorial Board D. R. Cox (University of Oxford) A. Agresti (University of Florida) B. Hambly (University of Oxford) S. Holmes (Stanford University) X.-L. Meng (Harvard University) IMS Monographs are concise research monographs of high quality on any branch of statistics or probability of sufficient interest to warrant publication as books. Some concern relatively traditional topics in need of up-to-date assessment. Others are on emerging themes. In all cases the objective is to provide a balanced view of the field. The Skew-Normal and Related Families ADELCHI AZZALINI Universita` degli Studi di Padova with the collaboration of ANTONELLA CAPITANIO Universita` di Bologna University Printing House, Cambridge CB2 8BS, United Kingdom Published in the United States of America by Cambridge University Press, New York Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107029279 © Adelchi Azzalini and Antonella Capitanio 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Azzalini, Adelchi, author. The skew-normal and related families / Adelchi Azzalini, Universita` degli Studi di Padova with the collaboration of Antonella Capitanio, Universita` di Bologna. pages cm Includes bibliographical references and index. ISBN 978-1-107-02927-9 (Hardback) 1. Distribution (Probability theory) I. Capitanio, Antonella, 1964– author. II. Title. QA273.6.A98 2014 ′ 519.2 4–dc23 2013030070 ISBN 978-1-107-02927-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page vii 1 Modulation of symmetric densities 1 1.1 Motivation 1 1.2 Modulation of symmetry 2 1.3 Some broader formulations 12 1.4 Complements 17 Problems 22 2 The skew-normal distribution: probability 24 2.1 The basic formulation 24 2.2 Extended skew-normal distribution 35 2.3 Historical and bibliographic notes 41 2.4 Some generalizations of the skew-normal family 46 2.5 Complements 50 Problems 54 3 The skew-normal distribution: statistics 57 3.1 Likelihood inference 57 3.2 Bayesian approach 82 3.3 Other statistical aspects 85 3.4 Connections with some application areas 89 3.5 Complements 93 Problems 94 4 Heavy and adaptive tails 95 4.1 Motivating remarks 95 4.2 Asymmetric Subbotin distribution 96 4.3 Skew-t distribution 101 4.4 Complements 119 Problems 123 v vi Contents 5 The multivariate skew-normal distribution 124 5.1 Introduction 124 5.2 Statistical aspects 142 5.3 Multivariate extended skew-normal distribution 149 5.4 Complements 160 Problems 165 6 Skew-elliptical distributions 168 6.1 Skew-elliptical distributions: general aspects 168 6.2 The multivariate skew-t distribution 176 6.3 Complements 187 Problems 193 7 Further extensions and other directions 196 7.1 Use of multiple latent variables 196 7.2 Flexible and semi-parametric formulation 203 7.3 Non-Euclidean spaces 208 7.4 Miscellanea 211 8 Application-oriented work 215 8.1 Mathematical tools 215 8.2 Extending standard statistical methods 218 8.3 Other data types 226 Appendix A Main symbols and notation 230 Appendix B Complements on the normal distribution 232 Appendix C Notions on likelihood inference 237 References 241 Index 256 Preface Since about the turn of the millennium, the study of parametric families of probability distributions has received new, intense interest. The present work is an account of one approach which has generated a great deal of activity. The distinctive feature of the construction to be discussed is to start from a symmetric density function and, by suitable modification of this, generate a set of non-symmetric distributions. The simplest effect of this process is represented by skewness in the distribution so obtained, and this explains why the prefix ‘skew’ recurs so often in this context. The focus of this con- struction is not, however, skewness as such, and we shall not discuss the quintessential nature of skewness and how to measure it. The target is in- stead to study flexible parametric families of continuous distributions for use in statistical work. A great deal of those in standard use are symmetric, when the sample space is unbounded. The aim here is to allow for pos- sible departure from symmetry to produce more flexible and more realistic families of distributions. The concentrated development of research in this area has attracted the interest of both scientists and practitioners, but often the variety of propos- als and the existence of related but different formulations bewilders them, as we have been told by a number of colleagues in recent years. The main aim of this work is to provide a key to enter this theme. Besides its role as an introductory text for the newcomer, we hope that the present book will also serve as a reference work for the specialist. This is not the first book covering this area: there exists a volume, edited by Marc Genton in 2004, which has been very beneficial to the dissemin- ation of these ideas, but since its publication many important results have appeared and the state of the art is now quite different. Even today a definit- ive stage of development of this field has not been reached, if one assumes for a moment that such a state can ever be achieved, but we feel that the material is now sufficiently mature to also be fruitfully used for routine work of non-specialists. vii viii Preface The general framework and the key concepts of our development are formulated in Chapter 1. Subsequent chapters develop specific directions, in the univariate and in the multivariate case, and discuss why other dir- ections are given lesser importance or even neglected. Some people may find it surprising that quite ample space is given to univariate distributions, considering that the context of multivariate distributions is where the new proposals appear more significant. However, besides its interest per se, the univariate case facilitates the exposition of many concepts, even when their main relevance is in the multivariate context. There is a noticeable difference in the more articulate expository style of Chapters 1 to 6 compared with the briefer – even meagre one might say – summaries employed in Chapters 7 and 8, which deal with more specific themes. One reason for this choice is the greater importance given to the exposition of the basic concepts, recalling our main target in writing the book, and certain applied topics do not require a detailed discussion after the foundations of the construction are in place. Moreover, some of the more specialized or advanced topics are still in an evolutionary state, and any attempt to arrange them in an organized system is likely to become obsolete quite rapidly. Chapters 1 to 6 are organized with a set of complements each, dealing with some more specialized topics. At first reading or if a reader is inter- ested in getting a grasp of the key concepts only, these complements can be skipped without hindrance to understanding the core parts. At the end of these chapters there are sets of problems of varied levels of difficulty. As a rule of thumb, the harder ones are those with a reference at the end. The development of this work has greatly benefited from the generous help of Giuliana Regoli, who has dedicated countless hours to examin- ing and discussing with us many mathematical aspects. Obviously, any re- maining errors are our own responsibility. We are also grateful to Elvezio Ronchetti, Marco Minozzo and Chris Adcock for comments on aspects of robustness, time series and quantitative finance, respectively, and to Marc Genton for several remarks on the nearly final draft. Even if in a less tan- gible form, our views on this research area have benefited from interac- tions with people of the ‘skew community’, with whom we have shared our enthusiasm during these years. It has been a stimulating and rewarding enterprise. Adelchi Azzalini and Antonella Capitanio February 2013