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**Correlation and dependence.**
*(English)*
Zbl 0977.62004

London: Imperial College Press (ISBN 1-86094-264-4/hbk; 978-1-86094-975-3/ebook). xiv, 219 p. (2001).

This monograph is an attempt to gather the results of the investigations of dependence properties, especially those which were made during the last thirty years. It is written for a graduate course; an initial background in mathematical statistics and probability theory and integral calculus is required. It starts with an introduction into independence and correlation sketching the different aspects and the historical development. It becomes quite clear that the well established correlation coefficent, the rank correlation and Kendall’s tau are still the most important measures. But there are many other approaches, too.

Concepts of dependence and stochastic ordering are the theme of chapter 3. A main part deals with different concepts of bivariate positive dependence and their generalisations for more than two variables. The notion of bivariate positive dependence is transferred also to positive dependence ordering.

Chapter 4, the largest chapter of the book, is devoted to copulas. Copulas provide a uniform way to represent bivariate distributions on the unit square. They allow to separate the effect of dependence from the effects of the marginal distributions.

Their properties are given and general methods for constructing copulas are presented. An essential one is due to L. Rüschendorf [Ann. Inst. Stat. Math. 37, 225-233 (1985; Zbl 0573.62045)], but many authors have made contributions, too. Special kinds of copulas are pursued in greater detail. It is often possible to generalize the method of constructing copulas for more than two variables, but there are compatibility constraints on the marginals. A particular case of Rüschendorfs construction method leads to the Farlie-Gumbel-Morgenstern models of dependence. These models are dealt with in chapter 5. Here, scattered results are collected and also some novel results are given.

Finally, the global measures of dependence arc confronted with concepts of local dependence between random variables. For that purpose, chapter 6 first reviews global measures of dependence. Then local indices of dependence are discussed.

Concepts of dependence and stochastic ordering are the theme of chapter 3. A main part deals with different concepts of bivariate positive dependence and their generalisations for more than two variables. The notion of bivariate positive dependence is transferred also to positive dependence ordering.

Chapter 4, the largest chapter of the book, is devoted to copulas. Copulas provide a uniform way to represent bivariate distributions on the unit square. They allow to separate the effect of dependence from the effects of the marginal distributions.

Their properties are given and general methods for constructing copulas are presented. An essential one is due to L. Rüschendorf [Ann. Inst. Stat. Math. 37, 225-233 (1985; Zbl 0573.62045)], but many authors have made contributions, too. Special kinds of copulas are pursued in greater detail. It is often possible to generalize the method of constructing copulas for more than two variables, but there are compatibility constraints on the marginals. A particular case of Rüschendorfs construction method leads to the Farlie-Gumbel-Morgenstern models of dependence. These models are dealt with in chapter 5. Here, scattered results are collected and also some novel results are given.

Finally, the global measures of dependence arc confronted with concepts of local dependence between random variables. For that purpose, chapter 6 first reviews global measures of dependence. Then local indices of dependence are discussed.

Reviewer: R.Schlittgen (Hamburg)

### MSC:

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

62H20 | Measures of association (correlation, canonical correlation, etc.) |

62E10 | Characterization and structure theory of statistical distributions |