Importance of Tail Dependence in Bivariate Frequency Analysis

Abstract

This paper highlights the importance of taking into account the tail dependence in the context of bivariate frequency analysis based on copulas. Three nonparametric estimators of the tail-dependence coefficient are compared by simulations with seven families of copulas. We choose the two estimators most adapted to a bivariate frequency analysis of the annual maximum flows and the corresponding flow hydrograph volumes of the Loire River ͑France͒. In this example, the bivariate return period and the conditional density of the volume given that the flow exceeds a given threshold are computed. The results show, as can be expected, that out of the seven copula families tested, five overestimate the return periods of correlated extreme events. These results bring to the forefront the importance of taking into account the tail dependence in order to estimate the risk adequately.

Upper tail dependence in rainfall extremes: would we know it if we saw it?

“…These works provide quite extensive simulation studies devised to assess the reliability of a set of parametric and nonparametric k U estimators as well as several conclusive warnings about their use in practical analyses. Poulin et al (2007) made an ad hoc Monte Carlo experiment to choose the most appropriate estimator for a specific case study, retaining the Coles-Heffernan-Tawn k CHT U estimator (Coles et al 1999) and Capéraà-Fougères-Genest k CFG U estimator (Capéraà et al 1997). Poulin et al (2007) also stressed the caveats previously reported by Schmidt (2003) and Frahm et al (2005).…”

“…Poulin et al (2007) made an ad hoc Monte Carlo experiment to choose the most appropriate estimator for a specific case study, retaining the Coles-Heffernan-Tawn k CHT U estimator (Coles et al 1999) and Capéraà-Fougères-Genest k CFG U estimator (Capéraà et al 1997). Poulin et al (2007) also stressed the caveats previously reported by Schmidt (2003) and Frahm et al (2005). Serinaldi (2008) exploited the relationship between k U and Kendall correlation coefficient s K to build a diagnostic plot useful for the model selection.…”

“…The use of k U in hydrological analyses can be dated back to the works of Poulin et al (2007) and Serinaldi (2008) concerning bivariate frequency analyses of the annual maximum flows and the corresponding flow hydrograph volumes, and the pairwise analysis of rainfall data at multiple locations, respectively. As for the introduction of copulas in hydrology, the theoretical apparatus used in those works was essentially borrowed from econometric literature, namely, Schmidt (2003), Frahm et al (2005) and Schmidt and Stadtmüller (2006).…”

“…In order to shed light on such a matter, we recall that the recommendations provided by Schmidt (2003), Schmidt and Stadtmüller (2006), Poulin et al (2007), and Frahm et al (2005) are based on simulation experiments in which a unique value of the overall correlation is used, namely s K ¼ 1=3 in Schmidt (2003) and Frahm et al (2005), Pearson correlation q P ¼ 0:25 in Schmidt and Stadtmüller (2006) and s K ¼ 0:51 in Poulin et al (2007). Based on these simulation settings, these studies conclude that:…”

Geostatistical interpolation using copulas

“…In Malevergne and Sornette [2003] the authors showed that the usual Gaussian type of dependence is not appropriate to estimate financial risks, since it underestimates the dependence of extremes. There are several hydrological applications, most of them related to the analysis of extremes [Salvadori et al, 2007;Favre et al, 2004;Poulin et al, 2007].…”