Stability and contagion measures for spatial extreme value analyses

As part of global climate change an accelerated hydrologic cycle (including an increase in heavy precipitation) is anticipated. So, it is of great importance to be able to quantify high-impact hydrologic relationships, for example, the impact that an extreme precipitation (or temperature) in a location has on a surrounding region. Building on the Multivariate Extreme Value Theory we propose a contagion index and a stability index. The contagion index makes it possible to quantify the effect that an exceedance above a high threshold can have on a region. The stability index reflects the expected number of crossings of a high threshold in a region associated to a specific location i, given the occurrence of at least one crossing at that location. We will find some relations with well-known extremal dependence measures found in the literature, which will provide immediate estimators. For these estimators an application to the annual maxima precipitation in Portuguese regions is presented.

[1]  Marta Ferreira,et al.  Extremal dependence: some contributions , 2011, 1108.1972.

[2]  Helena Ferreira,et al.  Dependence between two multivariate extremes , 2011 .

[3]  Regina Y. Liu,et al.  Extreme Value Theory Approach to Simultaneous Monitoring and Tresholding of Multiple Risk Indicators , 2006 .

[4]  Rafael Schmidt,et al.  Non‐parametric Estimation of Tail Dependence , 2006 .

[5]  Nicole A. Lazar,et al.  Statistics of Extremes: Theory and Applications , 2005, Technometrics.

[6]  Richard L. Smith,et al.  The behavior of multivariate maxima of moving maxima processes , 2004 .

[7]  J. Teugels,et al.  Statistics of Extremes: Theory and Applications , 2004 .

[8]  Jonathan A. Tawn,et al.  A dependence measure for multivariate and spatial extreme values: Properties and inference , 2003 .

[9]  Rafael Schmidt,et al.  Tail dependence for elliptically contoured distributions , 2002, Math. Methods Oper. Res..

[10]  Martin Schlather,et al.  Models for Stationary Max-Stable Random Fields , 2002 .

[11]  Kevin E. Trenberth,et al.  Conceptual Framework for Changes of Extremes of the Hydrological Cycle with Climate Change , 1999 .

[12]  Kevin E. Trenberth,et al.  Atmospheric Moisture Residence Times and Cycling: Implications for Rainfall Rates and Climate Change , 1998 .

[13]  Stuart G. Coles,et al.  Regional Modelling of Extreme Storms Via Max‐Stable Processes , 1993 .

[14]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[15]  L. de Haan,et al.  Stationary min-stable stochastic processes , 1986 .

[16]  L. de Haan,et al.  A Spectral Representation for Max-stable Processes , 1984 .

[17]  Masaaki Sibuya,et al.  Bivariate extreme statistics, I , 1960 .

[18]  C. Fonseca.  Generalized madogram and pairwise dependence of maxima over two disjoint regions of a random field , 2011 .

[19]  Haijun Li,et al.  Orthant tail dependence of multivariate extreme value distributions , 2009, J. Multivar. Anal..

[20]  Richard L. Smith,et al.  MAX-STABLE PROCESSES AND SPATIAL EXTREMES , 2005 .

[21]  Richard L. Smith,et al.  Characterization and Estimation of the Multivariate Extremal Index , 1996 .

[22]  Pirooz Vatan,et al.  Max-infinite divisibility and max-stability in infinite dimensions , 1985 .