Robustness of statistical tests. (English) Zbl 0699.62033

Kariya, Takeaki; Sinha, Bimal K.

Statistical Modeling and Decision Science. Boston, MA etc.: Academic Press, Inc. xv, 189 p. $ 44.95 (1989).

This book deals with the robustness of statistical tests in multivariate analysis and its application to a wide variety of multiparameter hypotheses testing problems. Statistical inference in multivariate analysis in general is based on the assumption that the observations follow multivariate normal distributions. Of course, this assumption considerably simplifies the underlying analysis, preserves the linear structure of the problem and often admits transformation groups under which the problems remain invariant. In practice, however, the assumption of normality is often very questionable. Hence, it is important to investigate the robustness of optimal tests derived under the model of normality against deviations from this model.
It is the purpose of this book to develop a general, systematic finite sample theory of robustness of tests and to apply this theory to (optimal) tests considered under normality. The robustness is the exact (not approximate) robustness in that sense, that the optimality of a test under normality holds exactly under non-normality. The basic concepts discussed here can be described as null robustness, non-null robustness which, respectively, deal with the null and non-null distributions of a test statistic and its optimality, such as uniformly most powerful, uniformly most powerful invariant and locally best invariant properties.
As non-normal distributions the authors focus on orthogonally invariant and elliptically symmetric distributions because these distributions preserve the invariant structure of problems, an important fact in multiparameter hypotheses testing problems, and also possess many similar distributional properties as the multivariate normal distribution. In addition, the class of these distributions is very broad and includes such heavy-tailed distributions as the multivariate Cauchy distribution, the t-distribution and the contaminated normal distribution.
Chapter 1 deals with elliptically symmetric distributions and left- orthogonally invariant distributions and some properties of these two families of distributions. In Chapter 2, the general invariance approach to testing is outlined. The theory of invariant measures on locally compact topological groups is discussed and then the notions of group actions, quotient space and homogeneous space are explained.
In chapter 3, the core chapter of this book, a general theory of null, nonnull and optimality robustness of a test is developed. Most of the multiparameter testing problems are listed and the invariance structure of these problems is discussed. In Chapter 4, the robustness of Student’s t-test and of tests for serial correlation is established in many different non-normal cases.
Chapter 5 describes in detail the general multivariate analysis of variance problem which includes as a special case the well-known multivariate analysis of variance problem. The robustness properties of the tests in these problems, in particular the robustness of Hotelling’s \(T^ 2\)-test and the ANOVA F-test, are established. Chapter 6 presents robust tests for a variety of covariance structures. It is shown that the standard normal-theory optimum invariant tests for independence, sphericity, and a specified covariance structure are robust for non- normal distributions.
In Chapter 7, the problem of the detection of outliers with respect to both mean slippage and dispersion slippage is investigated in detail. Chapter 8 deals with the comparison of two populations in terms of their location and scale parameters in the nonnormal and nonexponential case. There is, however, a serious lack of nullrobustness in all cases.

Reviewer: H.Büning

MSC:

62F35	Robustness and adaptive procedures (parametric inference)
62H15	Hypothesis testing in multivariate analysis
62A01	Foundations and philosophical topics in statistics
62-02	Research exposition (monographs, survey articles) pertaining to statistics

Keywords:

tests for sphericity; tests for serial correlation; two-population; problems; Hotelling T-square test; multiparameter hypotheses testing problems; transformation groups; finite sample theory of robustness of tests; non-normality; null robustness; non-null robustness; uniformly most powerful; uniformly most powerful invariant; locally best invariant properties; elliptically symmetric distributions; multivariate normal distribution; heavy-tailed distributions; multivariate Cauchy distribution; t-distribution; contaminated normal distribution; orthogonally invariant distributions; invariant measures on locally compact topological groups; optimality robustness; Student’s t-test; general multivariate analysis of variance; ANOVA F-test; covariance structures; tests for independence; detection of outliers; mean slippage; dispersion slippage; nonexponential case