Robustness of Statistical Tests

Robustness of Statistical Tests provides a general, systematic finite sample theory of the robustness of tests and covers the application of this theory to some important testing problems commonly considered under normality. This eight-chapter text focuses on the robustness that is concerned with the exact robustness in which the distributional or optimal property that a test carries under a normal distribution holds exactly under a nonnormal distribution.

Chapter 1 reviews the elliptically symmetric distributions and their properties, while Chapter 2 describes the representation theorem for the probability ration of a maximal invariant. Chapter 3 explores the basic concepts of three aspects of the robustness of tests, namely, null, nonnull, and optimality, as well as a theory providing methods to establish them. Chapter 4 discusses the applications of the general theory with the study of the robustness of the familiar Student’s r-test and tests for serial correlation. This chapter also deals with robustness without invariance. Chapter 5 looks into the most useful and widely applied problems in multivariate testing, including the GMANOVA (General Multivariate Analysis of Variance). Chapters 6 and 7 tackle the robust tests for covariance structures, such as sphericity and independence and provide a detailed description of univariate and multivariate outlier problems. Chapter 8 presents some new robustness results, which deal with inference in two population problems.

This book will prove useful to advance graduate mathematical statistics students.

Preface

Introduction

Chapter 1 Spherically Symmetric Distributions

1.1 Why Normal and Why Not Spherical?

1.2 Elliptically Symmetric Distributions

1.3 Left-Orthogonally Invariant Distributions

Exercises

Chapter 2 Invariance Approach to Testing

2.1 Invariant Measures on Groups

2.2 Invariant Measures on Homogeneous Spaces

2.3 A Review of the Theory of Testing of Hypotheses

2.4 Distribution of a Maximal Invariant

Appendix to Chapter 2, Section 4

Exercises

Chapter 3 General Approach to the Robustness of Tests

3.1 Null, Nonnull and Optimality Robustness

3.2 Outline of Testing Problems Under Normality

3.3 General Theory on Null Robustness

3.4 General Theory on Nonnull Robustness

3.5 General Approach to Optimality Robustness

Exercises

Chapter 4 Robustness of t-Test and Tests for Serial Correlation

4.1 Formulation of the Problem

4.2 One-Sided Testing Problems Without Invariance

4.3 Two-Sided Testing Problems Without Invariance

4.4 UMPI Property of t-Test

4.5 Tests on Serial Correlation Without Invariance

Exercises

Chapter 5 General Multivariate Analysis of Variance (GMANOVA)

5.1 Introduction

5.2 GMANOVA Model and Problem

5.3 MANOVA Problem

5.4 GMANOVA Problem

Appendix

Exercises

Chapter 6 Tests for Covariance Structures

6.1 Introduction

6.2 Testing ∑12 = 0

6.3 Testing Sphericity

6.4 Testing ∑ = I vs ∑ > I

Exercises

Chapter 7 Detection of Outliers

7.1 Introduction

7.2 Test for Mean Slippage

7.3 Test for Dispersion Slippage

Appendix

Exercises

Chapter 8 Two-Population Problems

8.1 Introduction

8.2 Test of Equality of Two Location Parameters—Nonnormal Case

8.3 Test of Equality of Two Location Parameters—Nonexponential Case

8.4 Test of Equality of Two Scale Parameters—Nonnormal Case

8.5 Test of Equality of Two Scale Parameters—Nonexponential Case

Exercises

References

Author Index

Subject Index