Robust statistical procedures: asymptotics and interrelations.

*(English)*Zbl 0862.62032
Wiley Series in Probability and Mathematical Statistics. New York, NY: John Wiley & Sons Ltd. xiv, 466 p. (1996).

This is a modern book on robustness theory and is written for specialists. The book is organized in two parts. The first part supplies the basic theory. The first chapter gives an introduction. Chapter 2 provides a review of basic aspects of robustness and convergence. Chapter 3 is devoted to the discussion of the core of robust estimation. As expected, the main classes are characterized: \(M,L,R\), Pitman, differentiable statistical functions. The motivations are presented within a common frame. Chapters 4-6 contain the asymptotic representations of \(L,M\) and \(R\) estimators. Bahadur representation plays the central role in the study. The results on uniform asymptotic linearity are used for obtaining simplified computational procedures. The introduction of rank scores for estimating scale and regression parameters is particularly important. In Chapter 7, relations among these estimators are investigated.

Part II contains 3 chapters. It supplies the inferential theory: point estimation, confidence sets, testing of hypothesis. In Chapter 8, point estimation is mainly given by sequential methods. Recursive and \(K\)-step estimators are presented. They shall diminish the use of computing time. From the asymptotic equivalence of \(L,M\) and \(R\) estimators for location and regression, the properties of robust adaptive estimation are studied uniformly. Robust confidence sets estimation is the theme of Chapter 9. The discussion of the role of robustness in the evaluation of intensive procedures, jacknife and bootstrap is remarkable. Chapter 10 is devoted to the interpretation of the robustness of tests based on robust estimation.

Each chapter finishes with a set of theoretical problems. A large bibliography is given at the end of the book. Some complicated proofs are given in the Appendix. The book is clearly written and self contained, though some misprints are present. Its publication is an essential contribution to the theme.

Part II contains 3 chapters. It supplies the inferential theory: point estimation, confidence sets, testing of hypothesis. In Chapter 8, point estimation is mainly given by sequential methods. Recursive and \(K\)-step estimators are presented. They shall diminish the use of computing time. From the asymptotic equivalence of \(L,M\) and \(R\) estimators for location and regression, the properties of robust adaptive estimation are studied uniformly. Robust confidence sets estimation is the theme of Chapter 9. The discussion of the role of robustness in the evaluation of intensive procedures, jacknife and bootstrap is remarkable. Chapter 10 is devoted to the interpretation of the robustness of tests based on robust estimation.

Each chapter finishes with a set of theoretical problems. A large bibliography is given at the end of the book. Some complicated proofs are given in the Appendix. The book is clearly written and self contained, though some misprints are present. Its publication is an essential contribution to the theme.

Reviewer: C.N.Bouza (Caracas)

##### MSC:

62F35 | Robustness and adaptive procedures (parametric inference) |

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

62F10 | Point estimation |

62F25 | Parametric tolerance and confidence regions |

62J05 | Linear regression; mixed models |