Multivariate Kernel Smoothing and Its Applications, by J.E. Chacón and T. Duong, provides a comprehensive and up-todate 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.

中文翻译：

---

多元核平滑及其应用。

多元核平滑及其应用，由 JE Chacón 和 T. Duong 提供，提供了多元密度估计的全面和最新介绍。这本书写得很好，内容丰富，涉及内核平滑的基础知识和高级主题。对于从事多元密度估计的研究人员来说，这是对书架的宝贵补充，对于数学背景较少的从业者来说，它是一个可访问的参考。此外，它似乎适合作为研究生级别的潜在教科书。第一章介绍了密度估计作为一种探索性数据分析工具。没有深入研究数学公式，它使用有趣的真实数据示例演示了密度估计在聚类和分类问题中的一些实际应用。此外，它提供了关于如何处理专着的建议。根据读者的数学训练和本书的预期用途，有些读者可能会选择跳过某些章节而不会忘记主要内容。第 2-4 章回顾了多元密度估计中的各种主题，重点是非参数核密度估计方法。具体而言，第 2 章讨论了非参数核密度估计中不同带宽矩阵结构的优缺点，并呈现渐近结果。第 3 章研究了常用的用于密度估计的带宽矩阵选择器，并从经验和理论上比较了它们的性能。第 4 章转向旨在更好地分析重尾或有界数据的修正密度估计器。本书的其余部分从核密度估计转移到相关主题。第 5 章将密度估计扩展到密度导数估计，第 6 章演示了密度和密度导数估计的实际应用。第7章和第8章包括补充主题，如密度差估计、分类中的密度估计、测量误差数据的密度估计等；并用 R 中的插图讨论计算算法。这本书的一个吸引人的特点是它的介绍。正如作者在书中所说，他们希望这本书对广大读者有用，包括数据分析师、本科生、研究生和统计研究人员。牢记这一意图，作者在每章开始时都会对一个主题进行一般性介绍，接下来是一些真实的数据插图及其数学特性的总结。因此，要求更高的数学推导在每章的末尾。这种深思熟虑的处理使读者只需较少的数学准备就可以阅读本书。然而，作者并没有牺牲内容的清晰性，也没有忽视数学的严谨性。这种巧妙的介绍使这本书独一无二，因为它迎合了不同的读者群。这本书的另一个亮点是它包含了更高级的主题，例如密度导数估计、密度脊估计、密度差估计等。据我所知，这些主题中的大多数都没有包含在关于内核平滑的其他书籍中。换句话说，Chacón 和 Duong 的书比市场上的其他选择更全面。它包罗万象，因此它使人们无需为内核平滑中的高级主题寻找额外的资源。它绝对是统计研究人员或从业人员在处理与密度估计相关的问题时的首选参考。此外，这本书包含了一系列有趣的例子。在每一章中，读者都可以找到相关的真实数据分析和可视化，展示了不同方法的实际应用。作者还创建了一个网站，提供本书中用于生成图形和执行数据分析的所有 R 脚本，可在 http://www.mvstat.net/mvksa/ 上找到。虽然作者针对的是本科生，但我发现这本书更适合研究生读者。后面几章的内容对本科生来说似乎具有挑战性，即使是高级学生。此外，我希望看到更多的案例研究和练习，以便更容易将文本作为课程教科书采用。总的来说，我很高兴能够阅读这本书。它写得很漂亮。作者提供了许多关于多元核平滑的宝贵见解，我发现这些见解很有帮助。我期待在我的书架上有一本，我毫不怀疑它会成为我未来的研究参考书。