Multivariate Kernel Smoothing and Its Applications, by J.E. Chacón and T. Duong, provides a comprehensive and up-to-date introduction of multivariate density estimation. The book is well-written and informative addressing the fundamentals as well as advanced topics in kernel smoothing. It is a valuable addition to the shelf for researchers working on multivariate density estimation, and an accessible reference for practitioners with less mathematical background. Further, it seems appropriate as a potential textbook at the post-graduate level.
The first chapter gives an introduction of density estimation as an exploratory data analysis tool. Without diving into mathematical formulations, it demonstrates some practical applications of density estimation in clustering and classification problems using interesting real data examples. In addition, it offers suggestions on how to proceed with the monograph. Depending on the readers’ mathematical training and intended use of the book, some readers may choose to skip certain chapters without losing track of the main content.
Chapters 2–4 review various topics in multivariate density estimation, with a focus on nonparametric kernel density estimation methods. Specifically, Chapter 2 discusses the pros and cons of different bandwidth matrix structures in nonparametric kernel density estimation and presents asymptotic results. Chapter 3 investigates commonly used bandwidth matrix selectors for density estimation, and compares their performance both empirically and theoretically. Chapter 4 turns to modified density estimators that aim at better analyzing heavy-tailed or bounded data.
The rest of the book moves from kernel density estimation to related topics. Chapter 5 extends density estimation to density derivative estimation and Chapter 6 demonstrates practical applications of density and density derivative estimation. Chapters 7 and 8 include supplementary topics, such as density difference estimation, density estimation in classification, and density estimation for data with measurement errors; and discuss computational algorithms with illustrations in R.
One appealing feature of the book is its presentation. As the authors stated in the book, they hoped that the book is useful for a broad audience, including data analysts, undergraduates, post-graduates, and statistics researchers. Keeping this intention in mind, the authors begin each chapter with a general introduction of a topic, followed by some real data illustrations and a summary of its mathematical properties. Hence, the more demanding mathematical derivations are at the end of each chapter. This thoughtful treatment makes the book accessible to readers with less mathematical preparation. However, the authors did not sacrifice the clarity of the content or neglect mathematical rigor. This clever presentation makes the book unique in the sense that it caters to a diverse readership.
Another sparkle of the book is its inclusion of more advanced topics, such as density derivative estimation, density ridge estimation, density difference estimation, among others. To the best of my knowledge, most of these topics are not included in other books on kernel smoothing. In other words, the book by Chacón and Duong is more comprehensive than other options on the market. It is all-inclusive so that it frees one from looking for additional resources for advanced topics in kernel smoothing. It is definitely a go-to reference for statistics researchers or practitioners while working with problems related to density estimation.
Moreover, the book contains a collection of interesting examples. In every chapter readers can find relevant real data analysis and visualizations that demonstrate practical applications of different methods. The authors have also created a website, making all the R scripts used to generate the figures and perform data analysis in the book available, which can be found at http://www.mvstat.net/mvksa/.
Though the authors target advanced undergraduates, I found the book more suitable for a post-graduate audience. The content in its later chapters seems challenging for undergraduates, even for advanced students. In addition, I would have liked to see more case studies and exercises so that the text could be more easily adopted as a course textbook.
Overall, it was a great joy for me to review this book. It was written beautifully. The authors offered many valuable insights on multivariate kernel smoothing, which I found helpful. I am looking forward to having a copy on my bookshelf and I have no doubt that it will be my research reference book in the future.
Wellesley College
Wellesley, MA
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