Introduction to High-Dimensional Statistics, Christophe Giraud. Chapman\& Hall/CRC Press, 2021, 364 pp., $72.00 hardcover, ISBN 978-0-367-71622-6.

A deep dive into the theoretical underpinnings of common high-dimensional statistical 20 techniques, Dr. Giraud’s Introduction to High-Dimensional Statistics is a good reference for those who wish to explore the mathematical foundations of state-of-the-art multivariate methods. The book covers a wide array of topics, from estimation bounds to multivariate regression and even clustering. In this 2nd edition, Dr. Giraud expands his work to include more recent advances and statistical methods. The book consists of 12 chapters, starting with a brief introduction to the complexities of conducting statistics in high dimensions. The book then proceeds similar to a standard statistical textbook, moving from properties of statistical estimators to statistical modeling, including regression and then other more advanced topics. Each chapter concludes with a set of exercises, many of which are portions of proofs from the chapter left for the reader. All that being said, do not let the title of the book fool you. By the author’s own admission, this is not an introduction on the same level as Hastie et al.’s Elements of Statistical Learning. Instead, the focus of this book is on the mathematical foundations of high-dimensional techniques, proving theorems regarding properties of estimators. I must confess this is not quite what I expected upon first look; one truly cannot judge a book by its cover. That is not to say that this book is lacking. It is impressive in its efficient, yet thorough, presentation of the theory. I especially appreciated how the author took time at the beginning to illustrate some of the strange behavior one encounters in very high dimensions. However, I did find it jarring that mathematical notation was very often used without much introduction. There is an appendix with notations at the end of the book, but I would’ve rather had a bit more interpretation within the text rather than having to flip back and forth. There were also a few typographical errors and partial omissions of formulas, though I can’t be sure if this was part of the text or bugs in the software I used to read the digital version. In summary, this book would certainly make for a good graduate level textbook in an advanced course on statistical methods. If you are willing to put the necessary time and investment into rigorously exploring the foundations of high-dimensional statistics, than you can hardly do better than this book.