Consider a \(d\)-dimensional Archimax copula \(C\) with the representation \[C_{\psi,\ell}\left(\mathbf{u} \right) = \psi\left[\ell\left\lbrace \phi\left(u_{1} \right),\dots,\phi\left(u_{d} \right) \right\rbrace \right], \] where \(\ell: \mathbb{R}_{+}^{d} \to \mathbb{R}_{+}\) is a \(d\)-variate stable tail dependence function and \(\psi : \left[ 0,\infty\right) \to \left[0,1 \right] \) is an Archimedean generator with inverse \(\phi\). The class of Archimax copulas includes both Archimedean and extreme-value copulas.
From authors’ summary: “This article develops semiparametric inference for Archimax copulas: a nonparametric estimator of \(\ell\) and a moment-based estimator of \(\psi\) assuming the latter belongs to a parametric family. Conditions under which \(\psi\) and \(\ell\) are identifiable are derived. The asymptotic behavior of the estimators is then established under broad regularity conditions; performance in small samples is assessed through a comprehensive simulation study. The Archimax copula model with the Clayton generator is then used to analyze monthly rainfall maxima at three stations in French Brittany. The model is seen to fit the data very well, both in the lower and in the upper tail. The nonparametric estimator of \(\ell\) reveals asymmetric extremal dependence between the stations, which reflects heavy precipitation patterns in the area. Technical proofs, simulation results and \(\mathsf{R}\) code are provided in the Online Supplement.”