Multivariate Dispersion, Central Regions and Depth: The Lift Zonoid Approach.(Book Reviews)(Book Review)

Multivariate Dispersion, Central Regions and Depth: The Lift Zonoid Approach.

Karl MOSLER. New York: Springer-Verlag, 2002. ISBN: 0-387-95412-0. x + 291 pp. $69.95 (P).

This book introduces lift zonoids and their applications to statistics, probability, and economics. The author and his collaborators have published extensively in this area of research, and their theory has been integrated into this book. Overall, the book is well-written, and starts with a good overview of the contents and goals.

The book presents a number of concepts, including the zonoid, that likely will be new to many statisticians (as they were to me). The lift zonoid is a representation of a multivariate probability distribution (more generally, a positive measure in Euclidean space). The volume of a lift zonoid reflects the dispersion of the distribution.

Theorem 2.17 has a characterization (or alternative definition) of the lift zonoid. For a univariate cumulative distribution function (cdf) F with finite expectation, the lift zonoid is the convex hull of the points (0, 0), (1, [integral] x d F (x)), ([[integral].sub.-[infinity].sup.y] d F (x), [[integral].sub.-[infinity].sup.y] x d F (x)), y [member of] R, and ([[integral].sub.y.sup.[infinity]] d F (x), [[integral].sub.y.sup.[infinity]] x d F (x)), y [member of] R. With this characterization, I was able to duplicate the figures in Figures 1.4-1.6 (pp. 11-12). For an empirical version of the lift zonoid with data, replace F with an empirical distribution [F.sub.n].

The definition of a lift zonoid for a multivariate cdf F has several equivalent forms, one of which involves a convex hull. The lift zonoid is partly motivated as a multivariate extension of the Lorenz curve, which is used by economists for displaying income distributions. In the univariate case, the boundaries of the convex hull are the Lorenz and reverse Lorenz curves.

The main statistical applications are zonoid trimmed region (an example of a multivariate central region), the zonoid depth (a notion of multivariate data depth), and the mean hyperplane depth (a measure of combinatorial dispersion useful for two-sample comparisons). A data depth is a multivariate version of a quantile for quantifying whether a data point lies near the center or the extreme or somewhere in between, and a set of multivariate central regions can be useful for detecting outliers in data. The set of [alpha]-trimmed regions for 0 [less than or equal to] [alpha] [less than or equal to] 1 is a family of nested regions, with [alpha] = 1 corresponding to a set with one point (the mean) and [alpha] = 0 corresponding to the entire (Euclidean) space.

With the data depth, one can do inference, such as two-sample tests for scale and for location/scale. This is the topic of Chapter 5, written by Rainer Dyckerhoff. Further tests can be constructed based on mean hyperplane depth, the topic of Chapter 6.

The book includes plenty of useful graphs to help the reader see the zonoids and trimmed regions in two and three dimensions. Some additions that perhaps would have been helpful include some graphs comparing the different bivariate central regions in Section 3.5, some examples with data to compare the different notions of data depths in Section 4.3, and an internet location for the software for computing the data depths and multivariate central regions, in addition to the overview sections on algorithms (Sec. 3.9, 4.5). The computational details are nontrivial and are in papers published in COMPSTAT conference proceedings.

In addition to ideas useful in statistics, the book provides results on stochastic comparisons with applications in probability, operations research, and economics. The lift zonoid (partial) ordering, based on set inclusion of lift zonoids, and the lift zonoid volume ordering are stochastic orderings among probability distributions leading to multivariate indices of dispersion and measures of economic disparity and concentration when variables have an economic context. Multivariate versions of Gini indices are obtained. Also, with fixed univariate margins, dependence orders are proposed.

The book has elegant mathematical theory. To follow the mathematical details, the reader needs to be familiar with basic terminology and results of convex analysis and measure theory. Those interested mainly in applications can still benefit from reading Chapter 1 and some sections of subsequent chapters.

Harry JOE

University of British Columbia