Copula parameter estimation using Blomqvist’s beta

Résumé

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 $O(n)$ operations, as opposed to $O(n^2)$ 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.