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**Tensor methods in statistics.**
*(English)*
Zbl 0732.62003

Monographs on Statistics and Applied Probability. London: Chapman and Hall. xvi, 285 p. $ 35.00 (1987).

This is an unusual book dealing with less familiar topics in statistics using the not-so-well-known tensor notation. There is, indeed, some elegance in using tensor notation; in some areas it is a necessity. The book will be useful in familiarizing statisticians with the methods of tensor calculus.

The first three chapters provide an introduction to the topics discussed in some detail in the remaining five chapters. The first chapter gives a brief introduction to the summation convention and tensor calculus with applications to some topics in matrix theory followed by further results and exercises. The second and third chapters deal with moments, cumulants, generalized cumulants and invariants. The advantages in working with cumulants rather than with moments are stated. Lattice theory is introduced for possible applications in statistics and probability theory.

Chapter 4 is devoted to a discussion of sample cumulants as developed by Fisher, Tukey and others. Some applications of cumulants are mentioned. Chapters 5 and 6 discuss Edgeworth expansions and saddle-point approximations which have become important tools in asymptotic theory of statistics.

Chapters 7 and 8 deal with inference problems based on the likelihood function and ancillary statistics. In particular, the large sample properties of the maximum likelihood estimates and the likelihood ratio statistic with the Bartlett factor are discussed. The need for conditional inference and the difficulties associated with it are emphasized. Asymptotic theory based on ancillary statistics is briefly discussed.

The book is a valuable addition to the modern literature on statistical inference although many topics such as the relative performances of the Wald statistic, Rao score statistic [the reviewer, Proc. Cambridge Philos. Soc. 44, 50–57 (1948; Zbl 0034.07503)], the likelihood ratio statistic, and applications of Edgeworth expansions are not fully discussed. The bibliographic notes, examples and numerous exercises at the end of each chapter make up for what is not covered in the discussion of various topics.

The first three chapters provide an introduction to the topics discussed in some detail in the remaining five chapters. The first chapter gives a brief introduction to the summation convention and tensor calculus with applications to some topics in matrix theory followed by further results and exercises. The second and third chapters deal with moments, cumulants, generalized cumulants and invariants. The advantages in working with cumulants rather than with moments are stated. Lattice theory is introduced for possible applications in statistics and probability theory.

Chapter 4 is devoted to a discussion of sample cumulants as developed by Fisher, Tukey and others. Some applications of cumulants are mentioned. Chapters 5 and 6 discuss Edgeworth expansions and saddle-point approximations which have become important tools in asymptotic theory of statistics.

Chapters 7 and 8 deal with inference problems based on the likelihood function and ancillary statistics. In particular, the large sample properties of the maximum likelihood estimates and the likelihood ratio statistic with the Bartlett factor are discussed. The need for conditional inference and the difficulties associated with it are emphasized. Asymptotic theory based on ancillary statistics is briefly discussed.

The book is a valuable addition to the modern literature on statistical inference although many topics such as the relative performances of the Wald statistic, Rao score statistic [the reviewer, Proc. Cambridge Philos. Soc. 44, 50–57 (1948; Zbl 0034.07503)], the likelihood ratio statistic, and applications of Edgeworth expansions are not fully discussed. The bibliographic notes, examples and numerous exercises at the end of each chapter make up for what is not covered in the discussion of various topics.

Reviewer: C. Radhakrishna Rao

### MSC:

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

62A01 | Foundations and philosophical topics in statistics |

15A69 | Multilinear algebra, tensor calculus |

53A45 | Differential geometric aspects in vector and tensor analysis |