Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
extreme value theory; Markov chains; autoregressive processes; tail dependence
extreme value theory; Markov chains; autoregressive processes; tail dependence
Summary:
In what concerns extreme values modeling, heavy tailed autoregressive processes defined with the minimum or maximum operator have proved to be good alternatives to classical linear ARMA with heavy tailed marginals (Davis and Resnick [8], Ferreira and Canto e Castro [13]). In this paper we present a complete characterization of the tail behavior of the autoregressive Pareto process known as Yeh-Arnold-Robertson Pareto(III) (Yeh et al. [32]). We shall see that it is quite similar to the first order max-autoregressive ARMAX, but has a more robust parameter estimation procedure, being therefore more attractive for modeling purposes. Consistency and asymptotic normality of the presented estimators will also be stated.
References:
[1] M. T. Alpuim: An extremal markovian sequence. J. Appl. Probab. 26 (1989), 219-232. DOI 10.2307/3214030 | MR 1000283 | Zbl 0677.60026
[2] B. C. Arnold: Pareto Distributions. International Cooperative Publishing House, Fairland 1983. MR 0751409
[3] B. C. Arnold: Pareto processes. In: Handbook of Statistics (D. N. Shanbhag and C. R. Rao, eds.), Elsevier Science B.V. 2001, Vol. 19. MR 1861718 | Zbl 0981.60036
[4] S. Asmussen: Applied Probability and Queues. John Wiley & Sons, Chichester 1987. MR 0889893 | Zbl 0624.60098
[5] L. Canto e Castro: Sobre a Teoria Assintótica de Extremos. Ph.D. Thesis, FCUL 1992.
[6] M. R. Chernick: A limit theorem for the maximum of autoregressive processes with uniform marginal distribution. Ann. Probab. 9 (1981), 145-149. DOI 10.1214/aop/1176994514 | MR 0606803
[7] M. R. Chernick, T. Hsing, W. P. McCormick: Calculating the extremal index for a class of stationary sequences. Adv. Probab. 23 (1991), 835-850. DOI 10.2307/1427679 | MR 1133731 | Zbl 0741.60042
[8] R. Davis, S. Resnick: Basic properties and prediction of max-ARMA processes. Adv. Appl. Probab. 21 (1989), 781-803. DOI 10.2307/1427767 | MR 1039628 | Zbl 0716.62098
[9] D. J. Daley, J. Haslett: A thermal energy storage process with controlled input. Adv. Appl. Probab. 14 (1982), 257-271. DOI 10.2307/1426520 | MR 0650122 | Zbl 0479.60097
[10] A. L. M. Dekkers, J. H. J. Einmahl, L. de Haan: A moment estimator for the index of an extreme value distribution. Ann. Statist. 17 (1989), 1833-1855. DOI 10.1214/aos/1176347397 | MR 1026315 | Zbl 0701.62029
[11] H. Drees: Extreme quantile estimation for dependent data with applications to finance. Bernoulli 9 (2003), 617-657. DOI 10.3150/bj/1066223272 | MR 1996273 | Zbl 1040.62077
[12] H. Ferreira: The upcrossings index and the extremal index. J. Appl. Probab. 43(4) (2006), 927-937. DOI 10.1239/jap/1165505198 | MR 2274627 | Zbl 1137.60024
[13] M. Ferreira, L. Canto e Castro: Modeling rare events through a $p$RARMAX process. J. Statist. Plann. Inference 140 (2010), 11, 3552-3566. DOI 10.1016/j.jspi.2010.05.024 | MR 2659877
[14] M. Ferreira, H. Ferreira: On extremal dependence: some contributions. (In press).
[15] I. S. Helland, T. S. Nilsen: On a general random exchange model. J. Appl. Probab. 13 (1976), 781-790. DOI 10.2307/3212533 | MR 0431437 | Zbl 0349.60066
[16] B. M. Hill: A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 (1975), 1163-1174. DOI 10.1214/aos/1176343247 | MR 0378204 | Zbl 0323.62033
[17] J. R. M. Hosking, J. R. Wallis: Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29 (1987), 339-349. DOI 10.1080/00401706.1987.10488243 | MR 0906643 | Zbl 0628.62019
[18] T. Hsing, J. Hüsler, M. R. Leadbetter: On the exceedance point process for a stationary sequence. Probab. Theory Related Fields 78 (1988), 97-112. DOI 10.1007/BF00718038 | MR 0940870 | Zbl 0619.60054
[19] J. Klotz: Statistical inference in Bernoulli trials with dependence. Ann. Statist. 1 (1973), 373-379. DOI 10.1214/aos/1176342377 | MR 0381103 | Zbl 0256.62029
[20] M. R. Leadbetter: On extreme values in stationary sequences. Z. Wahrsch. verw. Gebiete 28 (1974), 289-303. DOI 10.1007/BF00532947 | MR 0362465 | Zbl 0265.60019
[21] M. R. Leadbetter, G. Lindgren, H. Rootzén: Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York 1983. MR 0691492 | Zbl 0518.60021
[22] M. R. Leadbetter, S. Nandagopalan: On exceedance point processes for stationary sequences under mild oscillation restrictions. In: Extreme Value Theory (J. Hüsler and R.-D. Reiss, eds.), Springer-Verlag 1989, pp. 69-80. MR 0992049 | Zbl 0677.60040
[23] A. V. Lebedev: Statistical analysis of first-order MARMA processes. Mat. Zametki 83 (2008), 4, 552-558. MR 2431621 | Zbl 1152.62059
[24] A. Ledford, J. A. Tawn: Statistics for near independence in multivariate extreme values. Biometrika 83 (1996), 169-187. DOI 10.1093/biomet/83.1.169 | MR 1399163 | Zbl 0865.62040
[25] A. Ledford, J. A. Tawn: Modelling dependence within joint tail regions. J. Royal Statist. Soc. Ser. B 59 (1997), 475-499. DOI 10.1111/1467-9868.00080 | MR 1440592 | Zbl 0886.62063
[26] V. Pareto: Cours d'economie Politique. F. Rouge, Lausanne Vol. II., 1897.
[27] J. Pickands III: Statistical inference using extreme order statistics. Ann. Statist. 3 (1975), 119-131. DOI 10.1214/aos/1176343003 | MR 0423667
[28] S. Resnick, C. Stǎricǎ: Consistency of Hill's estimator for dependent data. J. Appl. Probab. 32 (1995), 139-167. DOI 10.2307/3214926 | MR 1316799
[29] S. Resnick, C. Stǎricǎ: Tail index estimation for dependent data. Ann. Appl. Probab. 8 (1998), 4, 1156-1183. DOI 10.1214/aoap/1028903376 | MR 1661160
[30] H. Rootzén, M. R. Leadbetter, L. de Haan: Tail and Quantile Estimation for Strongly Mixing Stationary Sequences. Technical Report, UNC Center for Stochastic Processes, 1990.
[31] R. L. Smith: Estimating tails of probability distributions. Ann. Statist. 15 (1987), 1174-1207. DOI 10.1214/aos/1176350499 | MR 0902252 | Zbl 0642.62022
[32] H. C. Yeh, B. C. Arnold, C. A. Robertson: Pareto processes. J. Appl. Probab. 25 (1988), 291-301. DOI 10.2307/3214437 | MR 0938193 | Zbl 0658.62101
[33] Z. Zhang, R. L. Smith: Modelling Financial Time Series Data as Moving Maxima Processes. Technical Report Dept. Stat. (Univ. North Carolina, Chapel Hill, NC, 2001); http://www.stat.unc.edu/faculty/rs/ papers/RLS_Papers.html.
Search
Partner of
