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MAX-MIN DEPENDENCE COEFFICIENTS FOR MULTIVARIATE EXTREME VALUE DISTRIBUTIONS
HELENA FERREIRA
Dedicated to Nazar´e, my teacher of Probability, matrix of inspiration and energy
Abstract. We measure the dependence among subvectors of a random vector with Multivariate Extreme Value distribution by using the ex- pected value of a range and relate this coe cient of dependence with the multivariate tail dependence and extremal coe cients. The introduced coe cient extends the concept of madogram for several locations and several regions. The results are illustrated with some usual distributions and applied to financial data.
1. Introduction
The dependence structure of a Multivariate Extreme Value (MEV) distri- bution is completely characterised by its dependence function (Resnick, 1987; Beirlant et al., 2004). Since this function cannot be easily inferred from data the dependence coe cients are useful, despite the fact that one coe cient cannot preserve all the information about this function.
The most popular of the dependence coe cients are those based on the tail dependence (Sybuya, 1960; Li, 2009). They summarize the probability of occurrence of extreme values for one or more random variables given that an- other(s) assumes extreme values too. For the MEV distributions the extremal coe cient (Tiago de Oliveira, 1962-63; Smith, 1990) is certainly a crucial and perhaps insurmountable tool when we have to summarize the dependence. For a d dimensional random vector we have 2d d extremal coe cients whose con- sistency properties are discussed in Schlather and Tawn (2002). For an overview of other dependence measures see, for instance, Joe (1997).
Accepted: 17 February 2015.
2010 Mathematics Subject Classification. 60G70.
Key words and phrases. Multivariate Extreme Value distribution, dependence, range. The work was supported by PEst-OE/MAT/UI0212/2014.
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