Inference for Archimax copulas. (English) Zbl 1450.62046

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.”


62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H12 Estimation in multivariate analysis
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62G32 Statistics of extreme values; tail inference
60G70 Extreme value theory; extremal stochastic processes
62P12 Applications of statistics to environmental and related topics


simsalapar; R
Full Text: DOI Euclid


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