Statistical inference procedures for bivariate Archimedean copulas. (English) Zbl 0785.62032

Genest, Christian; Rivest, Louis-Paul

J. Am. Stat. Assoc. 88, No. 423, 1034-1043 (1993).

Summary: A bivariate distribution function \(H(x,y)\) with marginals \(F(x)\) and \(G(y)\) is said to be generated by an Archimedean copula if it can be expressed in the form \[ H(x,y)=\varphi^{- 1}[\varphi\{F(x)\}+\varphi\{G(y)\}] \] for some convex, decreasing function \(\varphi\) defined on \((0,1]\) in such a way that \(\varphi(1)=0\). Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thélot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription.
This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates \(X\) and \(Y\) in the light of a random sample \((X_ 1,Y_ 1),\dots,(X_ n,Y_ n)\). The key to the estimation procedure is a one- dimensional empirical distribution function that can be constructed whether the uniform representation of \(X\) and \(Y\) is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendall’s tau statistic, is seen to be \(\sqrt n\)-consistent, and an explicit formula for its asymptotic variance is provided. This leads to a strategy for selecting the parametric family of Archimedean copulas that provides the best possible fit to a given set of data. To illustrate these procedures, a uranium exploration data set is reanalyzed. Although the presentation is restricted to problems involving a random sample from a bivariate distribution, extensions to situations involving multivariate or censored data could be envisaged.

MSC:

62G05	Nonparametric estimation
62G30	Order statistics; empirical distribution functions
62G07	Density estimation

Keywords:

empirical process; Frailty model; \(U\) statistic; Archimedean copula; bivariate distributions; dependence structure; empirical distribution; semiparametric estimator; decomposition of Kendall’s tau statistic; asymptotic variance; uranium exploration data set