# zbMATH — the first resource for mathematics

Testing statistical hypotheses. 2nd ed. (English) Zbl 0608.62020
Wiley Series in Probability and Mathematical Statistics. New York: John Wiley & Sons, Inc., xx, 600 p. £44.15 (1986).
This is a revised and somewhat expanded version of the well known first edition (1959; Zbl 0089.14102). The organization of material in both editions is based on optimality considerations. However, the current edition places a much stronger emphasis on robustness properties and gives more attention to simultaneous inference procedures and admissibility.
The book consists of 10 chapters. Chapter 1 discusses various fundamental concepts such as invariance, unbiasedness, maximum likelihood, complete classes and sufficient statistics. Chapter 2 is a review of probability and measure. Chapter 3 treats the theory of uniformly most powerful tests with applications to testing the mean and variance of a normal distribution. The three sections on sequential procedures in the first edition have been deleted as they have become outdated but no new material has been added in their place. Theory and applications of unbiasedness are discussed in Chapter 4. Alternative models for $$2\times 2$$ tables and three factor contingency tables are discussed in this chapter. These were not contained in the first edition. Chapter 5 consists of several applications including comparison of means and variances of 2 normal distributions, regression and permutation tests. There are sections on robustness, the effect of dependence, randomization models and such which were not included in the earlier edition. Chapter 6 treats the topic of invariance. It contains new material on admissibility and equivariant confidence sets.
So far we have a one to one correspondence between the 2 editions. But Chapter 7 of the first edition has been expanded and divided into two separate chapters. The current edition considers the general univariate linear hypotheses in Chapter 7 and multivariate linear hypotheses in Chapter 8. Chapter 7 contains much additional material on robustness against nonnormality and simultaneous confidence sets which were not discussed in the earlier edition. Chapter 8 discusses several applications of the multivariate linear hypotheses and $$\chi^ 2$$ tests. Chapter 9 is titled ”Minimax Principle” and includes topics such as comparing two approximate hypotheses, the Hunt-Stein theorem and most stringent tests.
Chapter 10 treats the important topic of conditional inference which was not included in the first edition. It contains a discussion of ancillary statistics, optimal conditional tests and relevant subsets.
Every chapter is supplemented with an annotated list of references and a rich collection of problems, many of which are outlines of solutions or introduction to further topics. The exposition is clear and sufficiently rigorous. Much of the book can be read without knowledge of measure theory although familiarity with it will be of much help. Topics not covered in this book, or covered minimally, include sequential analysis, Bayesian approaches to testing, optimum experimental design, multiple decision procedures and survey sampling. These topics certainly deserve separate treatment.
This book and the companion volume “Theory of Point Estimation.” (1983; Zbl 0522.62020) by the author will undoubtedly be the standard graduate level textbooks on statistical inference for several years to come.

##### MSC:
 62F03 Parametric hypothesis testing 62F05 Asymptotic properties of parametric tests 62-02 Research exposition (monographs, survey articles) pertaining to statistics 62G10 Nonparametric hypothesis testing 62F35 Robustness and adaptive procedures (parametric inference) 62H15 Hypothesis testing in multivariate analysis