Page 13 - Elements of Distribution Theory

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Preface

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 approxima-
tion of probability distributions and associated quantities such as moments and cumulants.
Although distribution theory plays a central role in the development of statistical method-
ology, 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,
withoutrequiringanextensivebackgroundinmathematics.Chapters1to4coverbasictopics
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 cov-
ers 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. For instance,
Watson’s lemma and Laplace’s method are general, useful tools for approximating the
integrals that arise in statistics, and orthogonal polynomials are used in areas ranging from
nonparametric function estimation to experimental design.
Chapters 11 to 14 cover large-sample approximations to probability distributions. Chap-
ter 11 covers the basic ideas of convergence in distribution and Chapter 12 contains several
versions of the central limit theorem. Chapter 13 considers the problem of approximating
the distribution of statistics that are more general than sample means, such as nonlin-
ear functions of sample means and U-statistics. Higher-order asymptotic approximations
such as Edgeworth series approximations and saddlepoint approximations are presented in
Chapter 14.
Ihave attempted to keep each chapter as self-contained as possible, but some dependen-
cies are inevitable. Chapter 1 and Sections 2.1–2.4, 3.1–3.2, and 4.1-4.4 contain core topics
that are used throughout the book; the material covered in these sections will most likely be

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