Multivariate Statistical Simulation concerns the computer generation of multivariate probability distributions. Generation is used in a broader context than solely algorithm development. An important aspect of generating multivariate probability distributions is what should be generated, as opposed merely to what could be generated. This viewpoint necessitates an examination of distributional properties and of the potential payoffs of including particular distributions in simulation studies. Since other available books document many of the mathematical properties of distributions (e.g. Johnson and Kotz, Distributions in Statistics: Continuous Multivariate Distributions ), a complementary approach is taken here. Generation algorithms are presented in tandem with many graphic aids (three-dimensional and contour plots) that highlight distributional properties from a unique perspective. These plots reveal features of distributions that rarely emerge from preliminary algebraic manipulations.
The primary beneficiary of this book is the researcher who is confronted with the task of designing and executing a simulation study that will employ continuous multivariate distributions. The prerequisite for the reader is a relentless curiosity as to the behavior of the method, estimator, test, or system under investigation when various multivariate distributions are assumed. The multivariate distributions presented in this text can serve as simulation drones to satiate the researcher’s curiosity.
--From the Preface
In my opinion, the above fairly describes the work being reviewed. The chapter headings indicate what is covered: (1) Introduction, (2) Univariate Distributions and Their Generation, (3) Multivariate Generation Techniques, (4) Multivariate Normal and Related Distributions, (5) Johnson’s Translation System, (6) Elliptically Contoured Distributions, (7) Circular, Spherical, and Related Distributions (8) Khintchine Distributions, (9) Multivariate Burr, Pareto, and Logistic Distributions, (10) Miscellaneous Distributions, and (11) Research Directions. There are no proofs or exercises. If you cannot find it in the book, you ought to be able to find it in one of the 158 references or 106 supplementary references.
Besides possessing the relentless curiosity mentioned in the preface, the reader should also have finished a standard college calculus sequence and a few semesters of mathematical statistics and probability theory in order to understand the book. However, to fully appreciate what is presented would probably take a much stronger grounding in these subjects and in real analysis, since the author is very good at putting references to rigorous mathematics unobtrusively into the qualifiers of clear English prose.
Except for the plots (many, various, and valuable), the book seems to present little that is new. The value of this compendium of results from the literature lies in the fact that the information is gathered together in one place, selected, and shrewdly evaluated by an evident expert in the business. Since I am not a specialist in the field, I find it hard to judge how another expert might have handled the material. Certainly the introductory material on univariate distributions, where I am somewhat more at home, is right on the money. There are no obvious errors.
While recipes are given for higher dimensions, the emphasis of the material is on bivariate distributions, which turn out to be quite complicated and various enough in their own right.
Although it is well written, I would judge the book to be impossible to read straight through because of its relentlessly thorough exposition of fact after fact about an enormous amount of material. Here is an example of the style from a discussion of the contaminated normal distribution:
With appropriate choice of p, &dgr;, and k, many commonly used statistical methods perform abominably with data generated from (4.3). Not surprisingly, the distribution in (4.3) is a popular choice for researchers who have devised robust statistical methods. Moreover, with five parameters present, many distributional shapes (including bimodal) are possible, some of which may even resemble actual data.
And there is the rub. In spite of the amount of information packed into it, the book conveys the impression of being the sketchiest of outlines of a vast and largely unexplored continent of mathematics, full of many strange and wonderful beasts. Their forms are so various it is hard to organize them into anything with a coherent main theme or story to tell, and the contact points with mundane reality remain problematic. If you need to venture into this realm, here is about as much as we know now: the sketch of the geography and the necessary explorer’s gear so that you can bring them--and yourself--back alive.