The Annals of Applied Probability

The extremal index of a higher-order stationary Markov chain

Seokhoon Yun

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Abstract

The paper presents a method of computing the extremal index of a real-valued, higher-order (kth-order, $k \geq 1$) stationary Markov chain ${X_n}$. The method is based on the assumption that the joint distribution of $k +1$ consecutive variables is in the domain of attraction of some multivariate extreme value distribution. We introduce limiting distributions of some rescaled stationary transition kernels, which are used to define a new $k -1$th-order Markov chain ${Y_n}$, say. Then, the kth-order Markov chain ${Z_n}$ defined by $Z_n = Y_1 + \dots + Y_n$ is used to derive a representation for the extremal index of ${X_n}$. We further establish convergence in distribution of multilevel exceedance point processes for ${X_n}$ in terms of ${Z_n}$. The representations for the extremal index and for quantities characterizing the distributional limits are well suited for Monte Carlo simulation.

Article information

Source
Ann. Appl. Probab., Volume 8, Number 2 (1998), 408-437.

Dates
First available in Project Euclid: 9 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1028903534

Digital Object Identifier
doi:10.1214/aoap/1028903534

Mathematical Reviews number (MathSciNet)
MR1624945

Zentralblatt MATH identifier
0942.60038

Subjects
Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G10: Stationary processes 60G55: Point processes

Keywords
Extremal index multivariate extreme value distributions exceedance point processes stationary Markov chains

Citation

Yun, Seokhoon. The extremal index of a higher-order stationary Markov chain. Ann. Appl. Probab. 8 (1998), no. 2, 408--437. doi:10.1214/aoap/1028903534. https://projecteuclid.org/euclid.aoap/1028903534


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References

  • Asmussen, S. (1987). Applied Probability and Queues. Wiley, Chichester.
  • Berman, S. M. (1964). Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35 502-516.
  • Billingsley, P. (1985). Probability and Measure, 2nd ed. Wiley, New York.
  • Chernick, M. R. (1981). A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Ann. Probab. 9 145-149.
  • Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. Roy. Statist. Soc. Ser. B 53 377-392.
  • Dekkers, A. L. M. and de Haan, L. (1989). On the estimation of the extreme-value index and large quantile estimation. Ann. Statist. 17 1795-1832.
  • Galambos, J. (1987). The Asy mptotic Theory of Extreme Order Statistics, 2nd ed. Krieger, Florida. (1st ed. published 1978 by Wiley, New York.)
  • Hsing, T. (1984). Point processes associated with extreme value theory. Ph.D. dissertation, Dept. Statistics, Univ. North Carolina, Chapel Hill.
  • Hsing, T. (1987). On the characterization of certain point processes. Stochastic Process. Appl. 26 297-316.
  • Hsing, T. (1991). Estimating the parameters of rare events. Stochastic Process. Appl. 37 117-139.
  • Hsing, T. (1993). Extremal index estimation for a weakly dependent stationary sequence. Ann. Statist. 21 2043-2071.
  • Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.
  • Leadbetter, M. R. (1974). On extreme values in stationary sequences. Z. Wahrsch. Verw. Gebiete 28 289-303.
  • Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete 65 291-306.
  • Leadbetter, M. R., Lindgren, G. and Rootz´en, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Loy nes, R. M. (1965). Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36 993-999.
  • Marshall, A. W. and Olkin, I. (1983). Domains of attraction of multivariate extreme value distributions. Ann. Probab. 11 168-177.
  • Nandagopalan, S. (1990). Multivariate extremes and estimation of the extremal index. Ph.D. dissertation, Dept. Statistics, Univ. North Carolina, Chapel Hill.
  • Nummelin, E. (1984). General Irreducible Markov Chains and Non-Negative Operators. Cambridge Univ. Press.
  • O'Brien, G. L. (1974). The maximum term of uniformly mixing stationary processes. Z. Wahrsch. Verw. Gebiete 30 57-63.
  • O'Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Probab. 15 281-291.
  • Perfekt, R. (1994). Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Probab. 4 529-548.
  • Perfekt, R. (1997). Extreme value theory for a class of Markov chains with values in d. Adv. in Appl. Probab. 29 138-164.
  • Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119-131.
  • Pickands, J. (1981). Multivariate extreme value distributions. In Proceedings of the 43rd Session of the International Statistical Institute, Buenos Aires, Argentina 2 859-878.
  • Resnick, S. I. (1987). Extreme Values, Point Processes and Regular Variation. Springer, New York.
  • Rootz´en, H. (1978). Extremes of moving averages of stable processes. Ann. Probab. 6 847-869.
  • Rootz´en, H. (1988). Maxima and exceedances of stationary Markov chains. Adv. in Appl. Probab. 20 371-390.
  • Smith, R. L. (1987). Estimating tails of probability distributions. Ann. Statist. 15 1174-1207. Smith, R. L. (1992a). The extremal index for a Markov chain. J. Appl. Probab. 29 37-45. Smith, R. L. (1992b). On an approximation formula for the extremal index in a Markov chain. Mimeo Series 2087, Institute of Statistics, Dept. Statistics, Univ. North Carolina, Chapel Hill.
  • Smith, R. L. and Weissman, I. (1994). Estimating the extremal index. J. Roy. Statist. Soc. Ser. B 56 515-528.
  • Smith, R. L., Tawn, J. A. and Coles, S. G. (1997). Markov chain models for threshold exceedances. Biometrika 84 249-268.
  • Tawn, J. A. (1988). Bivariate extreme value theory: models and estimation. Biometrika 75 397- 415.
  • Tawn, J. A. (1990). Modelling multivariate extreme value distributions. Biometrika 77 245-253.
  • Yun, S. (1994). Extremes and threshold exceedances in higher order Markov chains with applications to ground-level ozone. Ph.D. dissertation, Dept. Statistics, Univ. North Carolina, Chapel Hill.