Einmahl, John H. J.; de Haan, L.; Sinha, Ashoke Kumar Estimating the spectral measure of an extreme value distribution. (English) Zbl 0905.62051 Stochastic Processes Appl. 70, No. 2, 143-171 (1997). Let \((X_i, Y_i),\;i=1,\dots , n\) be i.i.d. random vectors with a bivariate distribution function \(F\in \mathcal D(G)\) (domain of attraction of a bivariate extreme value distribution \(G).\) The distribution function \(G\) is assumed to have non-degenerate marginal distributions, and it is characterized by extreme value indices \(\gamma _1, \gamma _2\) and by the spectral measure \(\varPhi .\) An estimator \(\hat \varPhi \) of \(\varPhi \) is constructed on the basis of the sample \((X_i, Y_i)\) and the consistency of \(\hat \varPhi \) (weak as well as strong) is proved. Moreover, the asymptotic normality of this estimator is established under additional conditions on the distribution function \(F\) and the extreme value indices \(\gamma _1, \gamma _2\). Reviewer: Zuzana Prášková (Praha) Cited in 31 Documents MSC: 62G20 Asymptotic properties of nonparametric inference 60F15 Strong limit theorems 62G30 Order statistics; empirical distribution functions 62M15 Inference from stochastic processes and spectral analysis 60F17 Functional limit theorems; invariance principles 60G70 Extreme value theory; extremal stochastic processes Keywords:extreme values; spectral measure; empirical processes; Vapnik-Cervonenkis class PDFBibTeX XMLCite \textit{J. H. J. Einmahl} et al., Stochastic Processes Appl. 70, No. 2, 143--171 (1997; Zbl 0905.62051) Full Text: DOI References: [1] Deheuvels, P., On the limiting behaviour of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett., 12, 429-439 (1991) · Zbl 0749.62033 [2] Deheuvels, P.; Tiago, de Oliveira J., On the non-parametric estimation of the bivariate extreme-value distributions, Statist. Probab. Lett., 8, 315-323 (1989) · Zbl 0712.62031 [3] Dekkers, A.; Einmahl, J.; de Haan, L., A moment estimator for the index of an extreme-value distribution, Ann. Statist., 17, 1833-1855 (1989) · Zbl 0701.62029 [4] Dudley, R., Universal Donsker classes and metric entropy, Ann. Probab., 15, 1306-1326 (1987) · Zbl 0631.60004 [5] Einmahl, J., Poisson and Gaussian approximation of weighted local empirical processes, Stochastic Processes and Their Applications, 70, 31-58 (1997) · Zbl 0911.60004 [6] Einmahl, J.; de Haan, L.; Xin, H., Estimating a multidimensional extreme-value distribution, J. Multivariate Anal., 47, 35-47 (1993) · Zbl 0778.62047 [7] Gaenssler, P., Empirical Processes, IMS Lecture Notes - Monograph Ser., 3 (1983), Hayward [8] de Haan, L.; Resnick, S., Limit theory for multidimensional sample extremes, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 40, 317-337 (1997) · Zbl 0375.60031 [9] de Haan, L.; Resnick, S., Estimating the limit distribution of multivariate extremes, Commun. Statist., 9, 275-309 (1993) · Zbl 0777.62036 [10] Resnick, S., Extreme Values, Regular Variation and Point Processes. Springer, New York.; Resnick, S., Extreme Values, Regular Variation and Point Processes. Springer, New York. · Zbl 0633.60001 [11] Sheehy, A.; Wellner, J., Uniform Donsker classes of functions, Ann. Probab., 10, 1983-2030 (1992) · Zbl 0763.60012 [12] Smith, R. L.; Tawn, J. A.; Yuen, H. K., Statistics of multivariate extremes, Internat. Statist. Rev., 58, 47-58 (1990) · Zbl 0715.62095 [13] Tawn, J. A., Bivariate extreme value theory: models and estimation, Biometrika, 75, 397-415 (1988) · Zbl 0653.62045 [14] Tawn, J. A., Modelling multivariate extreme value distributions, Biometrika, 77, 245-253 (1990) · Zbl 0716.62051 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.