Copula parameter estimation using Blomqvist’s beta

Genest, Christian; Carabarín-Aguirre, Alberto; Harvey, Fanny

Journal de la société française de statistique, Numéro spécial sur les copules, Tome 154 (2013) no. 1, pp. 5-24.

Les auteurs s’intéressent à l’inversion du beta de Blomqvist comme estimateur des moments du paramètre de dépendance réel d’un modèle de copule bivarié. Cet estimateur est obtenu en isolant le paramètre de la copule dans l’équation $\beta ={\beta }_n$, où ${\beta }_n$ est un estimateur de rangs de $\beta$ déduit d’un échantillon aléatoire de taille $n$. La performance asymptotique et à taille finie de cet estimateur est comparée à celle d’un estimateur analogue obtenu en inversant le tau de Kendall. Bien que les résultats montrent que ce dernier est plus efficace, le calcul de ${\beta }_n$ ne requiert que $O\left(n\right)$ opérations et non $O\left(n^2\right)$ comme pour l’estimation du tau de Kendall. Pour $n$ grand, l’inversion de $\beta$ fournit donc rapidement un estimateur sans biais et une bonne valeur initiale pour la maximisation de la vraisemblance canonique.

The authors consider the inversion of Blomqvist’s beta as a method-of-moments estimator for a real-valued dependence parameter in a bivariate copula model. This estimator results from solving the equation $\beta ={\beta }_n$ for the copula parameter, where ${\beta }_n$ is a rank-based estimate of $\beta$ derived from a random sample of size $n$. Small- and large-sample comparisons are made between this estimator and an analogous estimator based on the inversion of Kendall’s tau. While the results show that the latter is more efficient, the computation of ${\beta }_n$ requires only $O\left(n\right)$ operations, as opposed to $O\left(n^2\right)$ for the estimation of Kendall’s tau. Thus for large $n$, the inversion of $\beta$ quickly leads to an unbiased estimator and a good starting value for canonical likelihood maximization.

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