Robust statistics.

*(English)*Zbl 0536.62025
Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons. IX, 308 p. (1981).

To my knowledge, the present monograph is the first systematic, booklength exposition of robust statistics. It gives a good foundation in robustness, but more to the theoretical than to the applied statistician. The treatment is theoretical, but the stress is on concepts rather than on mathematical completeness.

The level of presentation is deliberately uneven: in some chapters single cases are treated with mathematical rigor; in others the results obtained in the simple cases are transferred by analogy to more complicated situations, where proofs are not always available. Also, selected numerical algorithms for computing robust estimates are described and, where possible, convergence proofs are given. The main point of this book is on estimation theory, only a few sections in Chapter 10 are related to test theory.

Chapter 1 gives a general introduction and overview on the different concepts of robustness: qualitative, quantitative, influence curve, sensitivity curve, a.o. Chapter 2 contains an account of the formal mathematical background behind qualitative and quantitative robustness, e.g. weak topology, Lévy and Prohorov metrics, Fréchet and Gâteaux derivatives. Chapter 3 introduces and discusses the three basic types of estimates: M-, L-, and R-estimates. Chapter 4 treats the asymptotic minimax theory for estimating a location parameter. Chapter 5 deals with scale estimates and Chapter 6 with multiparameter problems, in particular joint estimation of location and scale.

Chapter 7 considers the regression problem and investigates robust estimates of the parameters. Chapter 8 deals with robust covariance and correlation matrices, and Chapter 9 with the robustness of design. Chapter 10 gives exact finite sample results, especially in connection with some robust tests as mentioned above. Finally, in Chapter 11 miscellaneous topics are discussed.

The level of presentation is deliberately uneven: in some chapters single cases are treated with mathematical rigor; in others the results obtained in the simple cases are transferred by analogy to more complicated situations, where proofs are not always available. Also, selected numerical algorithms for computing robust estimates are described and, where possible, convergence proofs are given. The main point of this book is on estimation theory, only a few sections in Chapter 10 are related to test theory.

Chapter 1 gives a general introduction and overview on the different concepts of robustness: qualitative, quantitative, influence curve, sensitivity curve, a.o. Chapter 2 contains an account of the formal mathematical background behind qualitative and quantitative robustness, e.g. weak topology, Lévy and Prohorov metrics, Fréchet and Gâteaux derivatives. Chapter 3 introduces and discusses the three basic types of estimates: M-, L-, and R-estimates. Chapter 4 treats the asymptotic minimax theory for estimating a location parameter. Chapter 5 deals with scale estimates and Chapter 6 with multiparameter problems, in particular joint estimation of location and scale.

Chapter 7 considers the regression problem and investigates robust estimates of the parameters. Chapter 8 deals with robust covariance and correlation matrices, and Chapter 9 with the robustness of design. Chapter 10 gives exact finite sample results, especially in connection with some robust tests as mentioned above. Finally, in Chapter 11 miscellaneous topics are discussed.

Reviewer: H.Büning

##### MSC:

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

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

62F10 | Point estimation |