Distribution theory lies at the interface of probability and statistics. It is closely related to probability theory; however, it differs in its focus on the calculation and approximation of probability distributions and associated quantities such as moments and cumulants. Although distribution theory plays a central role in the development of statistical methodology, distribution theory itself does not deal with issues of statistical inference.
Many standard texts on mathematical statistics and statistical inference contain either a few chapters or an appendix on basic distribution theory. I have found that such treatments are generally too brief, often ignoring such important concepts as characteristic functions or cumulants. On the other hand, the discussion in books on probability theory is often too abstract for readers whose primary interest is in statistical methodology.
The purpose of this book is to provide a detailed introduction to the central results of distribution theory, in particular, those results needed to understand statistical methodology, without requiring an extensive background in mathematics. Chapters 1 to 4 cover basic topics such as random variables, distribution and density functions, expectation, conditioning, characteristic functions, moments, and cumulants. Chapter 5 covers parametric families of distributions, including exponential families, hierarchical models, and models with a group structure. Chapter 6 contains an introduction to stochastic processes.
Chapter 7 covers distribution theory for functions of random variables and Chapter 8 covers distribution theory associated with the normal distribution. Chapters 9 and 10 are more specialized, covering asymptotic approximations to integrals and orthogonal polynomials, respectively. Although these are classical topics in mathematics, they are often overlooked in statistics texts, despite the fact that the results are often used in statistics.