Abstract
At high levels, the asymptotic distribution of a stationary, regularly varying Markov chain is conveniently given by its tail process. The latter takes the form of a geometric random walk, the increment distribution depending on the sign of the process at the current state and on the flow of time, either forward or backward. Estimation of the tail process provides a nonparametric approach to analyze extreme values. A duality between the distributions of the forward and backward increments provides additional information that can be exploited in the construction of more efficient estimators. The large-sample distribution of such estimators is derived via empirical process theory for cluster functionals. Their finite-sample performance is evaluated via Monte Carlo simulations involving copula-based Markov models and solutions to stochastic recurrence equations. The estimators are applied to stock price data to study the absence or presence of symmetries in the succession of large gains and losses.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Process. Appl. 99(1), 95–115 (2002)
Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl. 119(4), 1055–1080 (2009)
Beare, B.K., Seo, J.: Time irreversible copula-based markov models. Econometric Theory FirstView, 1–38 (2014)
Bortot, P., Coles, S.: A sufficiency property arising from the characterization of extremes of markov chains. Bernoulli 6(1), 183–190 (2000)
Bowman, R.G., Iverson, D.: Short-run overreaction in the New Zealand stock market. Pac. Basin Financ. J. 6(5), 475–491 (1998)
Chen, X., Fan, Y.: Estimation of copula-based semiparametric time series models. J. Econ. 130(2), 307–335 (2006)
Chen, X., Wu, W.B., Yi, Y.: Efficient estimation of copula-based semiparametric markov models. Ann. Stat. 37(6B), 4214–4253 (2009)
Chen, Y.-T., Chou, R.Y., Kuan, C.-M.: Testing time reversibility without moment restrictions. J. Econ. 95(1), 199–218 (2000)
Chen, Y.-T., Kuan, C.-M.: Time irreversibility and EGARCH effects in US stock index returns. J. Appl. Econom. 17(5), 565–578 (2002)
Davis, R., Mikosch, T.: The extremogram: a correlogram for extreme events. Bernoulli 15(4), 977–1009 (2009)
Davis, R.A., Mikosch, T., Zhao, Y.: Measures of serial extremal dependence and their estimation. Stoch. Process. Appl. 123(7), 2575–2602 (2013)
De Bondt, W.F.M., Thaler, R.: Does the stock market overreact? J. Financ. 40(3), 793–805 (1985)
Dekkers, A., Einmahl, J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Stat. 17(4), 1833–1855 (1989)
Demarta, S., McNeil, A.J.: The t copula and related copulas. Int. Stat. Rev. 73(1), 111–129 (2005)
Doukhan, P.: Mixing. Properties and Examples. Springer, New York (1995)
Drees, H.: A general class of estimators of the extreme value index. J. Stat. Plan. Inf. 66(1), 95–112 (1998a)
Drees, H.: On smooth statistical tail functionals. Scand. J. Stat. 25(1), 187–210 (1998b)
Drees, H., Rootzén, H.: Limit theorems for empirical processes of cluster functionals. Ann. Stat. 38(4), 2145–2186 (2010)
Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1), 126–166 (1991)
Gudendorf, G., Segers, J.: Extreme-value copulas. In: Copula Theory and Its Applications, Lecture Notes in Statistics. Springer Berlin Heidelberg (2010)
Janßen, A., Drees, H.: A stochastic volatility model with flexible extremal dependence structure (2013). Available at arXiv:1310.4621
Janßen, A., Segers, J.: Markov tail chains. Advances in Applied Probability, forthcoming (2014)
Joe, H.: Families of min-stable multivariate exponential and multivariate extreme value distributions. Stat. Probab. Lett. 9(1), 75–81 (1990)
Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Mathematica 131(1), 207–248 (1973)
Kulik, R., Soulier, P.: Heavy tailed time series with extremal independence (2013). Available at arXiv:1307.1501
Larsson, M., Resnick, S.I.: Extremal dependence measure and extremogram: the regularly varying case. Extremes 15(2), 231–256 (2012)
Leadbetter, M.: Extremes and local dependence in stationary sequences. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 65(2), 291–306 (1983)
Ledford, A., Tawn, J.: Statistics for near independence in multivariate extreme values. Biometrika 83(1), 169–187 (1996)
Ledford, A., Tawn, J.: Diagnostics for dependence within time series extremes. J. R. Stat. Soc. Ser. B 65(2), 521–543 (2003)
Norli, A.: Short run stock overreaction: Evidence from Bursa Malaysia. J. Econ. Manag. 4(2), 319–333 (2010)
Perfekt, R.: Extreme value theory for a class of Markov chains with values in \(\mathbb {R}^{d}\). Adv. Appl. Probab. 29(1), 138–164 (1997)
Pratt, J.W.: On interchanging limits and integrals. Ann. Math. Stat. 31(1), 74–77 (1960)
Segers, J.: Multivariate regular variation of heavy-tailed markov chains. Technical report, Université catholique de Louvain (2007). Available at arXiv:math/0701411
Smith, R.: Estimating tails of probability distributions. Ann. Stat. 15(3), 1174–1207 (1987)
Smith, R.L.: The extremal index for a Markov chain. J. Appl. Probab. 29(1), 37–45 (1992)
Tawn, J.A.: Bivariate extreme value theory: Models and estimation. Biometrika 75(3), 397–415 (1988)
van der Vaart, A.W., Wellner, J.A.: Weak Convergence of Empirical Processes. Springer, New York (1996)
Yun, S.: The distributions of cluster functionals of extreme events in a dth-order markov chain. J. Appl. Probab. 37(1), 29–44 (2000)
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Drees, H., Segers, J. & Warchoł, M. Statistics for tail processes of Markov chains. Extremes 18, 369–402 (2015). https://doi.org/10.1007/s10687-015-0217-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-015-0217-1