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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-ThĂ©lot, 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
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