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

Let (U, V) be a random vector with U ≤ 0, V ≤ 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 ≥ -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) ↑ 0 as n → ∞.

[1]  R. Reiss.  Approximate Distributions of Order Statistics , 1989 .

[2]  Barry C. Arnold.  Parameter Estimation for a Multivariate Exponential Distribution , 1968 .

[3]  K. Ghoudi,et al.  Non-Parametric Estimation of the Limit Dependence Function of Multivariate Extremes , 1999 .

[4]  P. Hall,et al.  Distribution and dependence-function estimation for bivariate extreme-value distributions , 2000 .

[5]  R. Rao,et al.  Normal Approximation and Asymptotic Expansions , 1976 .

[6]  F. Marohn.  LOCAL ASYMPTOTIC NORMALITY OF TRUNCATION MODELS , 1999 .

[7]  Nonparametric Estimation of the Dependence Function in Bivariate Extreme Value Distributions , 2001 .

[8]  Efficient Estimation of the Canonical Dependence Function , 2003 .

[9]  Kilani Ghoudi,et al.  Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles , 1998 .

[10]  Intrinsic estimation of the dependence structure for bivariate extremes , 1989 .

[11]  J. Pfanzagl.  Parametric Statistical Theory , 1994 .

[12]  S. Resnick,et al.  Limit theory for multivariate sample extremes , 1977 .

[13]  Kanti V. Mardia,et al.  Families of Bivariate Distributions , 1970 .

[14]  Rolf-Dieter Reiss,et al.  Efficient estimators and LAN in canonical bivariate POT models , 2003 .

[15]  Paul Deheuvels,et al.  On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions , 1991 .

[16]  Janet E. Heffernan,et al.  Dependence Measures for Extreme Value Analyses , 1999 .

[17]  S. Csőrgő,et al.  Testing for exponential and Marshall-Olkin distributions , 1989 .

[18]  Christian Genest,et al.  A nonparametric estimation procedure for bivariate extreme value copulas , 1997 .

[19]  L. de Haan,et al.  Estimating the spectral measure of an extreme value distribution , 1997 .

[20]  S. Nadarajah.  Approximations for Bivariate Extreme Values , 2000 .

[21]  I. Olkin,et al.  A Multivariate Exponential Distribution , 1967 .

[22]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[23]  Jonathan A. Tawn,et al.  Statistics of Multivariate Extremes , 1990 .

[24]  J. D. T. Oliveira.  Statistical Decision for Bivariate Extremes , 1989 .

[25]  E. Ziegel.  Extreme Value Theory and Applications , 1994 .

[26]  J. Hüsler,et al.  Laws of Small Numbers: Extremes and Rare Events , 1994 .

[27]  S. Coles,et al.  An Introduction to Statistical Modeling of Extreme Values , 2001 .

[28]  A characterization of the rate of convergence in bivariate extreme value models , 2002 .

[29]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .