Introduction to the Theory of Statistical Inference

This book details the theoretical underpinnings of statistical inference. It is not an introduction to statistics; rather, it provides in-depth explanations, complete with proofs, of how statistics works. Along the way, the shortcomings of various standard methods become obvious, allowing the authors to discuss alternatives.

The book has several user-friendly aspects. One is the use of eight example data sets, briefly described in the introduction, to illustrate the theory throughout the text. This repeated use of the same examples allows readers to focus their energy on applying a theoretical point under discussion to a familiar example rather than having to first become acquainted with a new example. Another big help are the detailed solutions provided for the problems that appear at the end of each chapter. (The solutions fill 30 of the book’s 270 pages.) Also helpful: Theoretical or difficult material that can be skipped is marked with an asterisk.

Readers without much experience in unpacking mathematical notation should come prepared to gain such experience working through this book. There is a page in the introduction that provides glosses for some (mostly statistical) notation, but it does not include, for example, the tensor product, which appears as early as page 8 and then many times throughout the book. An undergraduate wanting to learn from this text should ideally come armed with not only a good statistics course behind her but also experience working with sets of various types (as would come from a solid probability course) and a willingness to unpack notation. This is not a text that is readily digestible if one’s only experience with statistics is, say, a typical first exposure to t-tests, regression, and the like, unless one also has significant “mathematical maturity” besides.

The ideal audience for this text is probably graduate students or practitioners for whom reading the notation will be second nature. They will be treated to a clear exposition of the theory of statistical inference, along with complete proofs and familiar examples. The text analyzes not just methods one learns in a first statistics course, but alternatives as well (e.g., robust methods for handling outliers). Each chapter is capped by a further reading section that is at once comprehensive and concise.

David A. Huckaby is an associate professor of mathematics at Angelo State University.