A Primer on Statistical Distributions
A Primer on Statistical Distributions
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DOI：
10.1198/jasa.2004.s341
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发表时间：
2004-06
影响因子：
3.7
通讯作者：
C. Anderson‐Cook
中科院分区：
文献类型：
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作者：
C. Anderson‐Cook
作者在序言中指出，“分布及其属性和相互关系在统计项目的大多数高年级本科生和研究生课程中发挥着非常重要的作用。”统计专业的学生在学习分布时经常遵循一条不相交且有些武断的路径——这里一点、那里一点，因为其他课程中的特定主题需要特定分布的信息。我们对各种分布的理解水平影响了我们对许多其他统计主题的理解，因此扩大学生对不同分布集合的关键特征的理解只能增强他们的整体统计背景。本书旨在为分布的发现和理解提供一个更加连续和有组织的结构。作者创作了一本书，汇集了有关单变量（离散和连续）和多个多元分布的各种信息。本书旨在作为单学期或单学期课程的教科书，对于任何统计学家的书架来说也是一本有价值的参考书。我认为将这本书用于本科生课程是非常雄心勃勃的，因为所需的数学水平超出了大多数典型的本科生的水平。然而，它适合统计学学生的硕士课程。本书分为四个主要部分。第一章讨论一般分布的一些预备知识。第 2-9 章介绍了八个单变量离散分布，每章一个。第 11-23 章讨论了 13 个单变量连续分布。第 10 章和第 24 章，在第二部分和第三部分的末尾，对一些被认为不值得单独章节的其他发行版进行了一些简短说明。最后，第 25 章至第 27 章介绍了多元分布和两个特定分布的一些背景。还有其他几本关于发行版的书籍。 Bury（1999）的这本书涵盖了工程师常用的分布，以不太复杂的数学水平编写，并且更加强调如何根据各种分布假设的数据以及常见工程应用的一些实际方面进行推论。相比之下，《统计分布入门》的理论性要强得多，对于大多数其他学科的研究人员来说可能很难阅读。与 Johnson、Kotz 和 Kemp（1992）以及 Johnson、Kotz 和 Balakrishnan（1994、1995 和 1997）的文本相比，这本书不太详细，主题之间的衔接更容易，并且更容易适合用作研究生统计课程的教科书。第一章介绍了本书其余部分所需的许多基础知识，包括符号、一些关键数理统计概念的概述以及用于总结分布的重要特征的讨论。第 1 部分（第 2-10 章）有单独的章节专门讨论以下每个离散分布：均匀分布、简并分布、伯努利分布、二项分布、几何分布、负二项分布、超几何分布和泊松分布。尽管每一章都有一些与各个分布相关的独特主题，但大多数章节都包含有关符号、矩、卷积、生成函数和特征函数、限制分布和分解的部分。第 10 章包含有关波利亚分布、帕斯卡分布和负超几何分布的一些非常简短的注释。第 2 部分（第 11-24 章）处理连续分布，遵循与第 1 部分类似的格式。这部分考虑的各个分布包括：均匀分布、柯西分布、三角分布、幂分布、帕累托分布、β 分布、反正弦分布、指数分布、拉普拉斯分布、伽马分布、极值分布、逻辑分布和正态分布。第 3 部分（第 25-27 章）重点关注多元分布，一章开始考虑一些附加符号、边际和条件分布细节以及一些极限定理。在各自的章节中介绍的个体分布是多元正态分布和狄利克雷分布。写作风格有时有些简洁，缺少一些动机来说明为什么特定结果在更广泛的学科背景中特别令人感兴趣。如果这本书用作教科书，那么教师需要将这种动机添加到文本材料中，以让学生了解结果，并帮助他们优先考虑正在阅读的内容的重要性。本书几乎不包含任何示例，材料的呈现侧重于为每个发行版提供尽可能多的相关结果，考虑到所讨论的发行版的雄心勃勃的集合，这可能是可以理解的。许多部分的末尾都会提供少量练习，难度从非常简单到更具挑战性不等。尽管这些可能对教师有所帮助，但它们的范围有限，并且并不总是代表某个部分中的所有材料。我发现分布的呈现顺序与预期相比发生了令人耳目一新的变化。通常，我们的学生会认为指数分布族是唯一真正重要的分布组。作者遵循了贯穿发行版收集的逻辑组织，并设法摆脱了呈现顺序的常见限制。总体而言，对于统计学家来说，这是一本很好的参考书，可以在他们的书架上轻松获得来自许多常见分布的许多相关且有用的结果。据我所知，这也是第一本适合专门讨论发行版的研究生课程的书。围绕本书构建一门有趣且相关的课程并不困难，这对我们的学生非常有益，有助于让分布之旅不再那么随意，更有可能促进他们全面发展成为成熟的统计学家。
The authors note in the Preface that “distributions and their properties and interrelationships assume a very important role in most upper-level undergraduate as well as graduate courses in the statistics program.” Frequently students of statistics follow a disjoint and somewhat arbitrary path in learning about distributions—a little here and a little there, as particular topics in other courses require pieces of information for a particular distribution. Our level of understanding about a variety of distributions colors our grasp of many other statistical topics, and hence broadening students’ grasp of key features of a diverse collection of distributions can only enhance their overall statistical background. This book seeks to give the discovery and understanding of distributions a more sequential and organized structure. The authors have created a book that brings together a wealth of information about a diverse collection of univariate (discrete and continuous) and several multivariate distributions. Designed to be a textbook for a single-semester or single-term course, this book would also be a valuable reference for any statistician’s bookshelf. I think it would be very ambitious to use this book for an undergraduate class, because the mathematical level required would be beyond most typical undergraduate students. However, it would be appropriate for a master’s-level course for statistics students. The book is organized into four major sections. The rst chapter discusses some preliminaries about distributions in general. Chapters 2–9 present eight univariate discrete distributions, one per chapter. Chapters 11–23 discuss 13 univariate continuous distributions. Chapters 10 and 24, at the end of the second and third parts, give some short notes on a few miscellaneous distributions that are judged unworthy of their own separate chapters. Finally, Chapters 25–27 present some background on multivariate distributions and two particular distributions. Several other books on distributions exist. The book by Bury (1999) covers distributions commonly used by engineers, written at a less-sophisticated mathematical level and with a greater emphasis on how to make inferences based on data assumed from various distributions and on some practical aspects of common engineering applications. In comparison, A Primer on Statistical Distributions is considerably more theoretical and would likely be dif cult reading for most researchers from other disciplines. Compared with the texts by Johnson, Kotz, and Kemp (1992) and Johnson, Kotz, and Balakrishnan (1994, 1995, and 1997), the book is less detailed, ows more easily between topics, and would adapt much more easily to being used for a textbook for teaching a graduatelevel statistics course. Chapter 1 introduces many of the basics required in the remainder of the book, including notation, an overview of some of key mathematical statistics notions, and a discussion of important characteristics used to summarize distributions. Part 1 (Chaps. 2–10) has separate chapters devoted to each of the following discrete distributions: uniform, degenerate, Bernoulli, binomial, geometric, negative binomial, hypergeometric, and Poisson. Although each chapter has some unique topics relevant to the individual distributions, most chapters include sections on notation, moments, convolutions, generating and characteristic functions, limiting distributions, and decompositions. Chapter 10 contains some very brief notes on the Polya, Pascal, and negative hypergeometric distributions. Part 2 (Chaps. 11–24), dealing with continuous distributions, follows a similar format as Part 1. The individual distributions considered in this part include: uniform, Cauchy, triangular, power, Pareto, beta, arcsine, exponential, Laplace, gamma, extreme-value, logistic, and Normal. Part 3 (Chaps. 25–27), concentrating on multivariate distributions, begins with a chapter considering some additional notation, marginal and conditional distribution details, and some limit theorems. The individual distributions presented in their own chapters are the multivariate normal and the Dirichlet. The writing style is somewhat terse at times, with some missing motivation about why particular results are of special interest in the broader context of the discipline. If this book were used as a textbook, then the instructor would need to add this motivation to the material in the text to keep the students tuned into the results, and to help them prioritize the importance of what they are reading. The book includes almost no examples, and the presentation of material focuses on presenting as many relevant results for each distribution as possible, which may be understandable given the ambitious collection of distributions discussed. A small number of exercises are presented at the end of many sections, ranging in dif culty from quite straightforward to considerably more challenging. Although these may be helpful for an instructor, they are of a limited scope and are not always representative of all of the material in a section. I found the order in which the distributions are presented a refreshing change from the expected. Frequently, our students are left with a notion that the exponential family of distributions is the only really important group of distributions. The authors have followed a logical organization that leads through the collection of distributions, and have managed to shake free from the common constraints of the order of presentation. Overall, this is a good reference book for statisticians to have available on their bookshelf, to have easy access to a number of relevant and useful results from a number of common distributions. It also is the rst book that I am aware of that would be suitable for a graduate-level course devoted to distributions. It would not be dif cult to build an interesting and relevant course around this book that would be highly bene cial to our students, to help make the journey through the distributions less haphazard and more likely to enhance their overall development into mature statisticians.