The extremal index of a higher-order stationary Markov chain. (English) Zbl 0942.60038

Yun, Seokhoon

Ann. Appl. Probab. 8, No. 2, 408-437 (1998).

The author presents a method of computing the so-called extremal index [see R. M. Loynes, Ann. Math. Stat. 36, 993-999 (1965; Zbl 0178.53201)] of a real-valued \(k\)th order, \(k\geq 1\), stationary Markov chain. 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. Four examples illustrate how the method works.

Reviewer: Marius Iosifescu (Bucureşti)

MSC:

60G70	Extreme value theory; extremal stochastic processes
60J05	Discrete-time Markov processes on general state spaces
60G10	Stationary stochastic processes
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)

Citations:

Zbl 0178.53201

References:

[1]	Asmussen, S. (1987). Applied Probability and Queues. Wiley, Chichester. · Zbl 0624.60098
[2]	Berman, S. M. (1964). Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35 502-516. · Zbl 0122.13503
[3]	Billingsley, P. (1985). Probability and Measure, 2nd ed. Wiley, New York. · Zbl 0822.60002
[4]	Chernick, M. R. (1981). A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Ann. Probab. 9 145-149. · Zbl 0453.60026
[5]	Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. Roy. Statist. Soc. Ser. B 53 377-392. JSTOR: · Zbl 0800.60020
[6]	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. · Zbl 0699.62028
[7]	Galambos, J. (1987). The Asy mptotic Theory of Extreme Order Statistics, 2nd ed. Krieger, Florida. (1st ed. published 1978 by Wiley, New York.) · Zbl 0634.62044
[8]	Hsing, T. (1984). Point processes associated with extreme value theory. Ph.D. dissertation, Dept. Statistics, Univ. North Carolina, Chapel Hill.
[9]	Hsing, T. (1987). On the characterization of certain point processes. Stochastic Process. Appl. 26 297-316. · Zbl 0645.60057
[10]	Hsing, T. (1991). Estimating the parameters of rare events. Stochastic Process. Appl. 37 117-139. · Zbl 0722.62021
[11]	Hsing, T. (1993). Extremal index estimation for a weakly dependent stationary sequence. Ann. Statist. 21 2043-2071. · Zbl 0797.62018
[12]	Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York. · Zbl 0248.62021
[13]	Leadbetter, M. R. (1974). On extreme values in stationary sequences. Z. Wahrsch. Verw. Gebiete 28 289-303. · Zbl 0265.60019
[14]	Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete 65 291-306. · Zbl 0506.60030
[15]	Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York. · Zbl 0518.60021
[16]	Loy nes, R. M. (1965). Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36 993-999. · Zbl 0178.53201
[17]	Marshall, A. W. and Olkin, I. (1983). Domains of attraction of multivariate extreme value distributions. Ann. Probab. 11 168-177. · Zbl 0508.60022
[18]	Nandagopalan, S. (1990). Multivariate extremes and estimation of the extremal index. Ph.D. dissertation, Dept. Statistics, Univ. North Carolina, Chapel Hill. · Zbl 0881.60050
[19]	Nummelin, E. (1984). General Irreducible Markov Chains and Non-Negative Operators. Cambridge Univ. Press. · Zbl 0551.60066
[20]	O’Brien, G. L. (1974). The maximum term of uniformly mixing stationary processes. Z. Wahrsch. Verw. Gebiete 30 57-63. · Zbl 0277.60020
[21]	O’Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Probab. 15 281-291. · Zbl 0619.60025
[22]	Perfekt, R. (1994). Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Probab. 4 529-548. · Zbl 0806.60041
[23]	Perfekt, R. (1997). Extreme value theory for a class of Markov chains with values in d. Adv. in Appl. Probab. 29 138-164. JSTOR: · Zbl 0882.60049
[24]	Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119-131. · Zbl 0312.62038
[25]	Pickands, J. (1981). Multivariate extreme value distributions. In Proceedings of the 43rd Session of the International Statistical Institute, Buenos Aires, Argentina 2 859-878. · Zbl 0518.62045
[26]	Resnick, S. I. (1987). Extreme Values, Point Processes and Regular Variation. Springer, New York. · Zbl 0633.60001
[27]	Rootzén, H. (1978). Extremes of moving averages of stable processes. Ann. Probab. 6 847-869. · Zbl 0394.60025
[28]	Rootzén, H. (1988). Maxima and exceedances of stationary Markov chains. Adv. in Appl. Probab. 20 371-390. JSTOR: · Zbl 0654.60023
[29]	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. · Zbl 0642.62022
[30]	Smith, R. L. and Weissman, I. (1994). Estimating the extremal index. J. Roy. Statist. Soc. Ser. B 56 515-528. JSTOR: · Zbl 0796.62084
[31]	Smith, R. L., Tawn, J. A. and Coles, S. G. (1997). Markov chain models for threshold exceedances. Biometrika 84 249-268. JSTOR: · Zbl 0891.60047
[32]	Tawn, J. A. (1988). Bivariate extreme value theory: models and estimation. Biometrika 75 397- 415. JSTOR: · Zbl 0653.62045
[33]	Tawn, J. A. (1990). Modelling multivariate extreme value distributions. Biometrika 77 245-253. · Zbl 0716.62051
[34]	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.