Multivariate multiparameter extreme value models and return periods: A copula approach

Abstract

[1] Multivariate extreme value models are a fundamental tool in order to assess potentially dangerous events. The target of this paper is two-fold. On the one hand we outline how, exploiting recent theoretical developments in the theory of copulas, new multivariate extreme value distributions can be easily constructed; in particular, we show how a suitable number of parameters can be introduced, a feature not shared by traditional extreme value models. On the other hand, we introduce a proper new definition of multivariate return period and show the differences with (and the advantages over) the definition presently used in literature. An illustration involving flood data is presented and discussed, and a generalization of the well-known multivariate logistic Gumbel model is also given.Citation: Salvadori, G., and C. De Michele (2010), Multivariate multiparameter extreme value models and return periods:

Stochastic simulation of precipitation-consistent daily reference evapotranspiration using vine copulas

“…Only very limited applications can be found in the literature with multivariate analysis of rainfall (Gräler 2014;Gyasi-Agyei and Melching 2012;Zhang and Singh 2007;Kao and Govindaraju 2008;Salvadori and De Michele 2006;Grimaldi and Serinaldi 2006), floods (Zhang and Singh 2014;Xiong et al 2014;Chen et al 2012;Serinaldi and Grimaldi 2007;Genest et al 2007;Salvadori and De Michele 2010) and droughts (Kao and Govindaraju 2010;Song and Singh 2010;Wong et al 2010).…”

Multivariate assessment of droughts: Frequency analysis and dynamic return period

“…Clearly, other models could be more indicated for describing the random behavior of data sets different from the one investigated here. An algorithm for simulatingĈ is outlined in Durante and Salvadori [2010] and Salvadori and De Michele [2010].…”

“…[25] Similarly, considering the duration D, the Weibull distribution is always selected as the best AIC model, over all the N R randomizations (see Figure 2 The ''Clayton,'' ''Gumbel,'' and ''Frank'' labels denote the corresponding families of Archimedean 2-copulas [Nelsen, 2006;Salvadori et al, 2007]; the labels ''Mix[AB]'' denote a convex mixture C of the two families A and B indicated (i.e., C ¼ A þ 1 À ð ÞB with 2 0; 1 ½ ); the labels ''X[AB]'' denote the Khoudraji-Liebscher extraparametrization C of the two families A and B indicated [Durante and Salvadori, 2010;Salvadori and De Michele, 2010].…”

“…[35] In the present case, the bivariate survival model selected is a versatile four parameters dependence structure described in Durante and Salvadori [2010] and Salvadori and De Michele [2010], belonging to the so-called extraparametrized Khoudraji-Liebscher's family (here labeled as ''XClaytonFrank''). The expression ofĈ iŝ…”

Multivariate return period calculation via survival functions

“…As prescribed in Genest and Favre [2007], several bivariate copula families (like e.g., the Clayton, Frank, Gumbel, Gaussian, t-Student, their convex mixtures, and the corresponding extraparameterized Khoudraji-Liebscher's versions [Durante and Salvadori, 2010;Salvadori and De Michele, 2010]) are fitted on the (normalized) survival ranks of the data. As a result, the following three-parameter convex mixtureĈ of the Clayton (A) and the Gumbel (B) copulas turns out to be appropriate according to both the (corrected) Akaike and the Bayesian information criteria :Ĉ u; ; ; ð Þ¼ A u; ð Þþ 1 À ð ÞB u; ð Þ, where the ranges of the parameters are !…”