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On the distribution of Pickands coordinates in bivariate EV and GP models

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  • Falk, Michael
  • Reiss, Rolf-Dieter

Abstract

Let (U,V) be a random vector with U[less-than-or-equals, slant]0, V[less-than-or-equals, slant]0. The random variables Z=V/(U+V), C=U+V are the Pickands coordinates of (U,V). They are a useful tool for the investigation of the tail behavior in bivariate peaks-over-threshold models in extreme value theory. We compute the distribution of (Z,C) among others under the assumption that the distribution function H of (U,V) is in a smooth neighborhood of a generalized Pareto distribution (GP) with uniform marginals. It turns out that if H is a GP, then Z and C are independent, conditional on C>c[greater-or-equal, slanted]-1. These results are used to derive approximations of the empirical point process of the exceedances (Zi,Ci) with Ci>c in an iid sample of size n. Local asymptotic normality is established for the approximating point process in a parametric model, where c=c(n)[short up arrow]0 as n-->[infinity].

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  • Falk, Michael & Reiss, Rolf-Dieter, 2005. "On the distribution of Pickands coordinates in bivariate EV and GP models," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 267-295, April.
    • Handle: RePEc:eee:jmvana:v:93:y:2005:i:2:p:267-295
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    References listed on IDEAS

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    1. Falk, Michael & Reiss, Rolf-Dieter, 2003. "Efficient estimators and LAN in canonical bivariate POT models," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 190-207, January.
    2. Deheuvels, Paul, 1991. "On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 429-439, November.
    3. Einmahl, John H.J. & de Haan, Laurens & Sinha, Ashoke Kumar, 1997. "Estimating the spectral measure of an extreme value distribution," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 143-171, October.
    4. de Oliveira, J. Tiago, 1989. "Intrinsic estimation of the dependence structure for bivariate extremes," Statistics & Probability Letters, Elsevier, vol. 8(3), pages 213-218, August.
    5. Marohn F., 1999. "Local Asymptotic Normality Of Truncation Models," Statistics & Risk Modeling, De Gruyter, vol. 17(3), pages 237-254, March.
    6. Falk, Michael & Reiss, Rolf Dieter, 2002. "A characterization of the rate of convergence in bivariate extreme value models," Statistics & Probability Letters, Elsevier, vol. 59(4), pages 341-351, October.
    7. Jiménez, Javier Rojo & Villa-Diharce, Enrique & Flores, Miguel, 2001. "Nonparametric Estimation of the Dependence Function in Bivariate Extreme Value Distributions," Journal of Multivariate Analysis, Elsevier, vol. 76(2), pages 159-191, February.
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    Cited by:

    1. Frick, Melanie & Reiss, Rolf-Dieter, 2009. "Expansions of multivariate Pickands densities and testing the tail dependence," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1168-1181, July.
    2. Di Bernardino, Elena & Maume-Deschamps, Véronique & Prieur, Clémentine, 2013. "Estimating a bivariate tail: A copula based approach," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 81-100.

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