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Strong convergence of multivariate maxima

Part of: Multivariate analysis Stochastic processes Limit theorems

Published online by Cambridge University Press:  04 May 2020

Michael Falk ,
Simone A. Padoan  and
Stefano Rizzelli
Michael Falk*
Affiliation:
University of Würzburg
Simone A. Padoan*
Affiliation:
Bocconi University of Milan
Stefano Rizzelli*
Affiliation:
École Polytechnique Fédérale de Lausanne
*
*Postal address: Chair of Mathematics VIII, Emil-Fischer-Str. 30, 97074 Würzburg, Germany. Email address: [email protected]
**Postal address: Department of Decision Sciences, via Roentgen, 1 20136 Milan, Italy. Email address: [email protected]
***Postal address: EPFL-SB-MATH-STAT, MA B1 507, Station 8, 1015 Lausanne, Switzerland. Email address: [email protected]
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Abstract

It is well known and readily seen that the maximum of n independent and uniformly on [0, 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalised Pareto copula. Sklar’s theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.

Keywords

maximastrong convergencetotal variationcopulageneralised Pareto copulaD-normmultivariate max-stable distributiondomain of attraction

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62H10: Distribution of statistics 60F15: Strong theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 314 - 331
Copyright
© Applied Probability Trust 2020

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References

Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications. John Wiley, Chichester.10.1002/0470012382CrossRefGoogle Scholar
Berghaus, B. and Bücher, A. (2018). Weak convergence of a pseudo maximum likelihood estimator for the extremal index. Ann. Statist. 46, 2307–2335.10.1214/17-AOS1621CrossRefGoogle Scholar
Berghaus, B., Bücher, A. and Dette, H. (2013). Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence. J. Soc. Française Statist. 154, 116–137.Google Scholar
Bücher, A. and Segers, J. (2014). Extreme value copula estimation based on block maxima of a multivariate stationary time series. Extremes 17, 495–528.10.1007/s10687-014-0195-8CrossRefGoogle Scholar
Bücher, A. and Segers, J. (2018). Maximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time series. Bernoulli 24, 1427–1462.10.3150/16-BEJ903CrossRefGoogle Scholar
Bücher, A., Volgushev, S. and Zou, N. (2019). On second order conditions in the multivariate block maxima and peak over threshold method. J. Multivar. Anal. 173, 604–619.10.1016/j.jmva.2019.04.011CrossRefGoogle Scholar
Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer, London.CrossRefGoogle Scholar
de Haan, L. and Peng, L. (1997). Rates of convergence for bivariate extremes. J. Multivar. Anal. 61, 195–230.CrossRefGoogle Scholar
Deheuvels, P. (1980). Non parametric tests of independence. In Statistique non Paramétrique Asymptotique, ed. J. P. Raoult, Springer, Berlin, pp. 95–107.10.1007/BFb0097426CrossRefGoogle Scholar
Dombry, C. (2015). Existence and consistency of the maximum likelihood estimators for the extreme value index within the block maxima framework. Bernoulli 21, 420–436.10.3150/13-BEJ573CrossRefGoogle Scholar
Dombry, C., Engelke, S. and Oesting, M. (2017). Bayesian inference for multivariate extreme value distributions. Electron. J. Statist., 11, 4813–4844.10.1214/17-EJS1367CrossRefGoogle Scholar
Falk, M. (2019). Multivariate Extreme Value Theory and D-Norms. Springer, New York.CrossRefGoogle Scholar
Falk, M., Hüsler, J. and Reiss, R.-D. (2011). Laws of Small Numbers: Extremes and Rare Events, 3rd edn. Birkhäuser, Basel.10.1007/978-3-0348-0009-9CrossRefGoogle Scholar
Falk, M., Padoan, S. A. and Wisheckel, F. (2019). Generalized Pareto copulas: a key to multivariate extremes. J. Multivar. Anal. 174, 104538.10.1016/j.jmva.2019.104538CrossRefGoogle Scholar
Ferreira, A. and de Haan, L. (2015). On the block maxima method in extreme value theory: PWM estimators. Ann. Statist. 43, 276–298.CrossRefGoogle Scholar
Ghosal, S. and van der Vaart, A. (2017). Fundamentals of Nonparametric Bayesian Inference. Cambridge University Press.CrossRefGoogle Scholar
Gudendorf, G. and Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. J. Statist. Planning Infer. 142, 3073–3085.CrossRefGoogle Scholar
Kaufmann, E. and Reiss, R.-D. (1993). Strong convergence of multivariate point processes of exceedances. Ann. Inst. Statist. Math. 45, 433–444.CrossRefGoogle Scholar
Kleijn, B. J. K. (2017). On the frequentist validity of Bayesian limits. Preprint, available at arXiv:1611.08444v3 .Google Scholar
McNeil, A. J. and Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and $\ell_1$ -norm symmetric distributions. Ann. Statist. 37, 3059–3097.CrossRefGoogle Scholar
Marcon, G., Padoan, S. A., Naveau, P., Muliere, P. and Segers, J. (2017). Multivariate nonparametric estimation of the Pickands dependence function using Bernstein polynomials. J. Statist. Planning Infer. 183, 1–17.10.1016/j.jspi.2016.10.004CrossRefGoogle Scholar
Mhalla, L., Chavez-Demoulin, V. and Naveau, P. (2017). Non-linear models for extremal dependence. J. Multivar. Anal. 159, 49–66.10.1016/j.jmva.2017.04.006CrossRefGoogle Scholar
Padoan, S. A. and Rizzelli, S. (2019). Strong consistency of nonparametric Bayesian inferential methods for multivariate max-stable distributions. Preprint, available at arXiv:1904.00245v2 .Google Scholar
Reiss, R.-D. (1989). Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics. Springer, New York.CrossRefGoogle Scholar
Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York.Google Scholar
Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231.Google Scholar
Sklar, A. (1996) Random variables, distribution functions, and copulas – a personal look backward and forward. In Distributions with Fixed Marginals and Related Topics, eds L. Rüschendorf, B. Schweizer and M. D. Taylor. Lecture Notes – Monograph Series, Vol. 28. Institute of Mathematical Statistics, Hayward, CA, pp. 1–14.Google Scholar
Wong, W. H. and Shen, X. (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLES. Ann. Statist. 23, 339–362.10.1214/aos/1176324524CrossRefGoogle Scholar
Wellner, J. A. (1992). Empirical processes in action: a review. Internat. Statist. Rev. 60, 247–269.CrossRefGoogle Scholar