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**Quadratic forms in random variables. Theory and applications.**
*(English)*
Zbl 0792.62045

Statistics: Textbooks and Monographs. 126. New York: Marcel Dekker. xix, 367 p. (1992).

This book is devoted to quadratic forms in random variables, more specifically to quadratic forms and second degree polynomials in real normal random vectors and matrices. The area has grown considerably, since it plays a central technical role in several fields of statistics.

The book starts with two short introductory chapters of mathematical preliminaries. Then chapters 3 to 5 deal with quadratic forms in random vectors: Chapter 3 considers alternative representations, moments, cumulants and their corresponding generating functions, and geometrical interpretations; Chapter 4 deals with distributional problems, both exact and approximate; Cahpter 5 deals with situations involving the chi-square distribution, and questions of independence. This kind of elements is then studied in Chapter 6 for the case of random matrices, where the distributional problems are associated with the Wishart distribution. Finally, Chapter 7 (“Applications”) presents 19 statistical problems where quadratic forms are used.

The approach is formal, in the definition-lemma-theorem format, except Chapter 7, where some discussion of the applications is provided. In the Preface the authors state that “most of the results included in this monograph are proved; in some cases an outline of the proof and a reference containing the complete proof are provided. Many of the proofs require a good understanding of basic statistical principles as well as a thorough background in mathematics.”

In view of the preceding comments, the following claim of the authors is valid: “this book is developed as a reference book for theoretical as well as applied statisticians. Those belonging to the latter group will find the examples very useful.” The authors claim that the book “can also be used as a textbook for a one semester graduate course on quadratic forms in random variables for students specializing in mathematical statistics or multivariate analysis.” This claim will have to be considered with care by those interested in organizing such a course, because the approach implies that the topics appear detached from the statistical problems in which they arise.

Chapters 3, 4 and 5 contain many worked examples, and together with Chapter 6 contain a total of 61 exercises, some of which present further results. The bibliography contains in the order of 400 entries.

In summary, the authors have surveyed an important area of mathematics statistics, and provided a systematic collection of traditional and new results. The book is important as a reference. Perhaps it can be used in teaching, as a supplement to more standard textbooks on mathematical statistics or multivariate analysis. The level of operational mathematics is considerable, and the adopted format emphasizes the mathematical proofs.

The book starts with two short introductory chapters of mathematical preliminaries. Then chapters 3 to 5 deal with quadratic forms in random vectors: Chapter 3 considers alternative representations, moments, cumulants and their corresponding generating functions, and geometrical interpretations; Chapter 4 deals with distributional problems, both exact and approximate; Cahpter 5 deals with situations involving the chi-square distribution, and questions of independence. This kind of elements is then studied in Chapter 6 for the case of random matrices, where the distributional problems are associated with the Wishart distribution. Finally, Chapter 7 (“Applications”) presents 19 statistical problems where quadratic forms are used.

The approach is formal, in the definition-lemma-theorem format, except Chapter 7, where some discussion of the applications is provided. In the Preface the authors state that “most of the results included in this monograph are proved; in some cases an outline of the proof and a reference containing the complete proof are provided. Many of the proofs require a good understanding of basic statistical principles as well as a thorough background in mathematics.”

In view of the preceding comments, the following claim of the authors is valid: “this book is developed as a reference book for theoretical as well as applied statisticians. Those belonging to the latter group will find the examples very useful.” The authors claim that the book “can also be used as a textbook for a one semester graduate course on quadratic forms in random variables for students specializing in mathematical statistics or multivariate analysis.” This claim will have to be considered with care by those interested in organizing such a course, because the approach implies that the topics appear detached from the statistical problems in which they arise.

Chapters 3, 4 and 5 contain many worked examples, and together with Chapter 6 contain a total of 61 exercises, some of which present further results. The bibliography contains in the order of 400 entries.

In summary, the authors have surveyed an important area of mathematics statistics, and provided a systematic collection of traditional and new results. The book is important as a reference. Perhaps it can be used in teaching, as a supplement to more standard textbooks on mathematical statistics or multivariate analysis. The level of operational mathematics is considerable, and the adopted format emphasizes the mathematical proofs.

Reviewer: R.Mentz (S.M.de Tucuman)

### MSC:

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

62H10 | Multivariate distribution of statistics |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

15B52 | Random matrices (algebraic aspects) |

62E20 | Asymptotic distribution theory in statistics |