The skew-normal and related families. With the collaboration of Antonella Capitanio. (Q2871234)

It is curious to see that, according to Google, there are more than 800,000 items with key words `skew-normal distribution'. Everything started near 30 years ago after \textit{A. Azzalini} published a paper on this topic [Scand. J. Stat. 12, 171--178 (1985; Zbl 0581.62014)]. A part of the motivations came from applied statistics after many researchers realized that the classical normal distribution is not a good model for describing phenomena with one or another kind of skewness of the data. For sure Azzalini was also attracted by the opportunity to perform a detailed probabilistic analysis of this new class of distributions by establishing a series of beautiful theoretical results with a clear indication of possible directions for further studies. The papers by Azzalini and his co-authors were so well written that they immediately attracted the attention of probabilists and applied statisticians. The flow of works became soon visible. The class of skew normals was extended in several directions to cover the skewness phenomenon for many other distributions and also for multivariate distributions.NEWLINENEWLINEIt was natural to assume that one day Professor Azzalini, as a leading expert, will decide to prepare and publish a book. Fortunately, this day arrived, and we can enjoy today an excellent and well written book on all aspects: theory, applications, computations. The book was prepared together with Professor A. Capitanio in her rôle as a special collaborator.NEWLINENEWLINEThe materials are well structured and divided into 8 chapters: 1. Modulation of symmetric densities. 2. The skew-normal distribution: probability. 3. The skew-normal distribution: statistics. 4. Heavy and adaptive tails. 5. The multivariate skew-normal distribution. 6. Skew-elliptic distributions. 7. Further extentions and other directions. 8. Application-oriented work. Appendix A: Main symbols and notations. Appendix B: Complements on the normal distribution. Appendix C: Notions on likelihood inference. At the very end, there is a list of about 330 well selected references and an index.NEWLINENEWLINEOf special interest are the theoretical considerations. When reading the book we learn about ideas, techniques and non-trivial results. They are followed by illustrations, computations and graphs. This reviewer liked especially those illustrations involving wine data. What is rare for a book in statistics, at the end of each section there are theoretical problems/exercises. Solutions and hints are not provided, however, a serious reader, with some efforts, would be able to master the details. Any section of the book contains a part called `Historical and bibliographical notes'. It is so interesting to learn about all the stages in developing the notion of skewness for the normal and many other distributions. Also complements are given in many places showing further perspectives in both theory and applications.NEWLINENEWLINEIt is well-known that Azzalini maintains a popular web-site of papers in this area. Meanwhile, statistical software, R code, the package `sn', was developed and is available to users for years. The last update is very recent, it is dated 22 December 2014.NEWLINENEWLINEWho will benefit from this book? Everybody! Important is that the skew-distributions, normal and others, one-dimensional and multidimensional, became an essential part of modern statistical theory. Not less important are the numerous potential applications. Professor Azzalini and his collaborators deserve our thanks for their great contribution in this scientific area and in particular for writing the present book.NEWLINENEWLINEThe book can be strongly recommended to researchers in mathematical statistics and applied statisticians. Moreover, the book can be used as a source for special university graduate courses addressed to Master and/or PhD students.