Abstract
Let (X,Y) be a random vector which follows in its upper tail a bivariate extreme value distribution with reverse exponentialmargins. We show that the conditional distribution function (df) of X + Y, given that X + Y>c, converges to the df F (t) = t 2, \(t \in [0,1]\), as \(c\uparrow 0\) if and only if X,Y are tail independent. Otherwise, the limit is F (t) = t. This is utilized to test for the tail independence of X, Y via various tests, including the one suggested by the Neyman–Pearson lemma. Simulations show that the Neyman–Pearson test performs best if the threshold c is close to 0, whereas otherwise it is the Kolmogorov–Smirnov test that performs best. The mathematical conditions are studied under which the Neyman–Pearson approach actually controls the type I error. Our considerations are extended to extreme value distributions in arbitrary dimensions as well as to distributions which are in a differentiable spectral neighborhood of an extreme value distribution.
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References
Coles S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics, Springer, New York
Coles S.G., Heffernan J.E., Tawn J.A. (1999). Dependence measures for extreme value analyses. Extremes 2:339–365
Dupuis D.J., Tawn J.A. (2001). Effects of mis–specification in bivariate extreme value problems. Extremes 4:315–330
Falk M., Reiss R.-D. (2005). On Pickands coordinates in arbitrary dimensions. Journal of Multivariate Analysis 92: 426–453
Feller W. (1971). An Introduction to Probability Theory and its Applications, Vol II, 2nd ed. John Wiley, New York
Fuller W.A. (1976). Introduction to Statistical Time Series. John Wiley, New York
Galambos J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd ed. Krieger, Malabar
Geffroy J. (1958). Contribution à la théorie des valeurs extrêmes. Publications de l’Institut de Statistique de l’Université de Paris 7:37–121
Geffroy J. (1959). Contribution à la théorie des valeurs extrêmes, II. Publications de l’Institut de Statistique de l’Université de Paris 8:3–65
Ghoudi K., Khoudraji A., Rivest L.P. (1998). Statistical properties of couples of bivariate extreme–value copulas. Canadian Journal of Statistics 26:187–197
Gumbel E.J. (1960). Bivariate exponential distribution. Journal of the American Statistical Association 55:698–707
Heffernan J.E. (2000). A directory of coefficients of tail dependence. Extremes 3:279–290
Hüsler J., Reiss R.–D. (1989). Maxima of normal random vectors: Between independence and complete dependence. Statistics and Probability Letters 7:283–286
Johnson N.L., Kotz S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York
Kotz S., Nadarajah S. (2000). Extreme Value Distributions. Imperial College Press, London
Ledford A., Tawn J.A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83:169–187
Ledford A., Tawn J.A. (1997). Modelling dependence within joint tail regions. Journal of the Royal Statistical Society B 59:475–499
Ledford A., Tawn J.A. (1998). Concomitant tail behaviour for extremes. Advances in Applied Probability 30:197–215
Mardia K.V. (1970). Families of Bivariate Distributions. Griffin, London
Marshall A.W. and Olkin I. (1976). A multivariate exponential distribution. Journal of the American Statistical Association 62:30–44
Peng L. (1999). Estimation of the coefficients of tail dependence in bivariate extremes. Statistics and Probability Letters 43:399–409
Pickands III J. (1981). Multivariate extreme value distributions. In: Proceedings of the 43th Session of the International Statistical Institute (Buenos Aires), pp 859–878.
Reiss R.–D. (1993). A Course on Point Processes. Springer Series in Statistics. Springer, New York
Reiss R.–D., Thomas M. (2001). Statistical Analysis of Extreme Values, 2nd ed. Birkhäuser, Basel
Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes, applied Probability, Vol.~4. New York: Springer.
Sheskin D.J. (2004). Parametric and Nonparametric Statistical Procedures, 3rd ed. Chapmann & Hall, Boca Raton
Sibuya M. (1960). Bivariate extreme statistics. Annals of the Institute of Statistical Mathematics 11:195–210
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Falk, M., Michel, R. Testing for Tail Independence in Extreme Value models. Ann Inst Stat Math 58, 261–290 (2006). https://doi.org/10.1007/s10463-005-0016-6
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DOI: https://doi.org/10.1007/s10463-005-0016-6