Generalized Additive Models: An Introduction With R

also includes some topics one may not expect to find in a text on robust statistics, such as sequential M estimates and L estimates, multivariate admissibility and shrinkage, and goodness-of-fit tests. These are topics of special interest to the authors and to which they have made significant research contributions. The presentation of the theoretical material in the text is well organized and easy to follow. Chapters 1, 2, 3, and 6 provide a fairly good introduction to the theory of robust statistics. In particular, the discussion of statistical functionals and differentiability, along with the accompanying examples, offers a clear and accessible account of these challenging topics. There are some minor shortcomings in the theoretical presentation. The definition of a linear functional, which is used extensively in Chapter 1, is presumed to be known. The proof given on page 17 that shows that a Fréchet differentiable functional is also Gâteaux differentiable presumes the Gâteaux derivative is a linear functional, but this is the key to the proof. The definitions given for the maximum bias and the asymptotic maximum bias on pages 33 and 35, respectively, are redundant. On page 46, the breakdown point of an M estimate of location is given as 1/2 whenever the corresponding ψ function is odd and bounded, but the condition that the ψ function should also be monotonic is omitted. Expressions for the breakdown points of redescending M estimates of location are more complicated; see Huber (1984). There are some minor typographical errors, none of which should confuse a student reader except perhaps the one appearing on page 48 [where E(IF(x; X̃,P ))2 should read E(IF(x; X̃,P )2)]. The methodological and computational parts of the book are less extensive than the theoretical part and, consequently, are not as clearly written. The methods tend to lack sufficient motivation or explanation. For example, an illustration of robust regression with MM estimates, computed using R, is given at the end of Chapter 4. Within the chapter, though, the only mention of MM estimates is a one line general definition. No motivation or discussion of properties of MM estimates is given. In particular, the text does not note that they are high breakdown point estimates introduced to improve upon the poor normal efficiency of the S estimates of regression. Rather, on page 110, one finds the statement that the S estimates “combine the high breakdown point. . . with a fairly good efficiency at the normal model.” Aside from the use of R, the text does not discuss computational algorithms for robust methods. The examples using R are not always sufficiently documented and are sometimes simply presented as R commands with their corresponding outputs. As an example, the illustrations of robust regression using M and MM estimates in Chapter 4 do not indicate which M estimate or MM estimate is being computed (i.e., what the default is) or how the estimates are tuned. Nevertheless, the authors should be commended for assembling a ready reference of various R commands that are applicable to robust statistics. The book is intended to be a graduate level text. In the American system, it would not be appropriate for a Master’s level course in statistics or for a course that emphasizes methodology and applications, especially one that has students from other disciplines. Ideally, the text is more appropriate for a Ph.D. level research course in robust statistics, either as the primary text or as a secondary resource. The book tends to be overly ambitious for its size in trying to cover robust topics from in-depth theory to applications and illustrations with R programs. This results in a rather terse style, and so the text reads for the most part like lecture notes. This could be an advantage if used as a primary text in that it allows instructors to give their own motivation, explanations, and emphasis to the various topics. Provided an instructor does so, the book would provide a good basis for a research level course on robust statistics. At the very least, for those with an interest in robust statistics, this text is worth having in one’s personal library. If I were teaching a graduate course in robust statistics, I would seriously consider this text. It has the unfortunate timing, however, of being published at the same time as the long awaited text by Maronna, Martin, and Yohai (2006). Although less theoretical, this latter text tends to be more comprehensive and more contemporary in its coverage of robust statistics.