##
**An introduction to copulas. Properties and applications.**
*(English)*
Zbl 0909.62052

Lecture Notes in Statistics (Springer). 139. New York, NY: Springer. xi, 216 p. (1999).

The notion of a copula was introduced by A. Sklar [see Kybernetika, Praha 9, 449-460 (1973; Zbl 0292.60036)] to describe the connection between a multivariate distribution function and its marginal distribution functions. Up to continuity properties it is a distribution function with uniform marginals. Sklar’s representation theorem identifies the \(n\)-dimensional distribution functions \(H\) with marginals \(F_1,\dots, F_n\) as \(C(F_1(x_1),\dots, F_n(x_n))\) for a copula \(C\), also called sometimes dependence function.

Sharp upper and lower bounds for copulas are given by the Hoeffding-Fréchet bounds and various construction methods are discussed in this volume. Of particular interest are shuffles of \(M\) which correspond to completely dependent random variables. An interesting fact is that the class of shuffles of \(M\) is dense in the class of all copulas and so any copula (in particular the product copula) can be uniformly approximated by completely dependent shuffles of \(M\). A simple and useful class of copulas are archimedian copulas of the form \(C(u,v)= \varphi^{-1} (\varphi(u)+ \varphi(v))\), \(\varphi\) convex strictly decreasing.

The notion of a copula is scale invariant; it therefore captures scale invariant dependence properties of distributions. In this sense many nonparametric measures of dependence and nonparametric statistical procedures are based on the distance between a copula to the independence copula. In this respect also questions of optimal couplings of distributions are touched as originating in early work of the Italian school of probabilists.

This book can be considered as a systematic continuation of previous work on the modelling of multivariate statistical distributions. It discusses useful construction methods for modelling statistical dependence and introduces several important aspects of analysis of dependence notions. It is not encyclopaedic in nature but gives an excellent introduction to some basic questions in statistical modelling.

Sharp upper and lower bounds for copulas are given by the Hoeffding-Fréchet bounds and various construction methods are discussed in this volume. Of particular interest are shuffles of \(M\) which correspond to completely dependent random variables. An interesting fact is that the class of shuffles of \(M\) is dense in the class of all copulas and so any copula (in particular the product copula) can be uniformly approximated by completely dependent shuffles of \(M\). A simple and useful class of copulas are archimedian copulas of the form \(C(u,v)= \varphi^{-1} (\varphi(u)+ \varphi(v))\), \(\varphi\) convex strictly decreasing.

The notion of a copula is scale invariant; it therefore captures scale invariant dependence properties of distributions. In this sense many nonparametric measures of dependence and nonparametric statistical procedures are based on the distance between a copula to the independence copula. In this respect also questions of optimal couplings of distributions are touched as originating in early work of the Italian school of probabilists.

This book can be considered as a systematic continuation of previous work on the modelling of multivariate statistical distributions. It discusses useful construction methods for modelling statistical dependence and introduces several important aspects of analysis of dependence notions. It is not encyclopaedic in nature but gives an excellent introduction to some basic questions in statistical modelling.

Reviewer: L.Rüschendorf (Freiburg i.Br.)

### MSC:

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

60A05 | Axioms; other general questions in probability |

62H99 | Multivariate analysis |

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

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |