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

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