Geodesic
On Tail Dependence: A Characterization for First-Order Max-Autoregressive Processes
Matematičeskie zametki, Tome 90 (2011) no. 6, pp. 902-917.

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In this paper, we consider first-order MARMA or ARMAX processes and a modified version of these involving a power transformation, denoted pARMAX. We assume Pareto-type tails, the most interesting case for inference within these processes. Some well-known dependence measures of multivariate extreme value theory are considered in a time series framework. In calculating these measures, we find that ARMAX and pARMAX have opposite behavior in concomitant extremes, covering all types of tail dependence. This characterization will serve modeling purposes.
Keywords: extreme value theory, max-autoregressive processes, tail dependence, ARMAX process.
Mots-clés : Markov chains 
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M. Ferreira. On Tail Dependence: A Characterization for First-Order Max-Autoregressive Processes. Matematičeskie zametki, Tome 90 (2011) no. 6, pp. 902-917. http://geodesic.mathdoc.fr/item/MZM_2011_90_6_a7/

[1] P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling Extremal Events. For Insurance and Finance, Appl. Math. (N. Y.), 33, Springer-Verlag, Berlin, 1997 | MR | Zbl

[2] R. A. Davis, S. I. Resnick, “Basic properties and prediction of max-ARMA processes”, Adv. in Appl. Probab., 21:4 (1989), 781–803 | DOI | MR | Zbl

[3] M. T. Alpuim, “An extremal Markovian sequence”, J. Appl. Probab., 26:2 (1989), 219–232 | DOI | MR | Zbl

[4] L. Canto e Castro, Sobre a Teoria Assintótica de Extremos, Ph. D. Thesis, The Faculty of Sciences of the University of Lisbon, 1992

[5] M. A. Ancona-Navarrete, J. A. Tawn, “A comparison of methods for estimating the extremal index”, Extremes, 3:1 (2000), 5–38 | DOI | MR | Zbl

[6] A. V. Lebedev, “Statisticheskii analiz MARMA-protsessov pervogo poryadka”, Matem. zametki, 83:4 (2008), 552–558 | MR | Zbl

[7] D. J. Daley, J. Haslett, “A thermal energy storage with controlled input”, Adv. in Appl. Probab., 14:2 (1982), 257–271 | DOI | MR | Zbl

[8] I. S. Helland, T. S. Nilsen, “On a general random exchange model”, J. Appl. Probab., 13:4 (1976), 781–790 | DOI | MR | Zbl

[9] Z. Zhang, R. L. Smith, Modeling Financial Time Series Data as Moving Maxima Processes, Preprint, The University of North Carolina at Chapel Hill, 2001

[10] P. Bortot, J. A. Tawn, “Models for the extremes of Markov Chains”, Biometrika, 85:4 (1998), 851–867 | DOI | MR | Zbl

[11] M. Sibuya, “Bivariate extreme statistics. I”, Ann. Inst. Statist. Math. Tokyo, 11:2 (1960), 195–210 | DOI | MR | Zbl

[12] A. W. Ledford, J. A. Tawn, “Statistics for near independence in multivariate extreme values”, Biometrika, 83:1 (1996), 169–187 | DOI | MR | Zbl

[13] A. W. Ledford, J. A. Tawn, “Modelling Dependence within joint tail regions”, J. Roy. Statist. Soc. Ser. B, 59:2 (1997), 475–499 | DOI | MR | Zbl

[14] G. Draisma, H. Drees, A. Ferreira, L. de Haan, “Bivariate tail estimation: dependence in asymptotic independence”, Bernoulli, 10:2 (2004), 251–280 | DOI | MR | Zbl

[15] M. Ferreira, L. Canto e Castro, “Tail and dependence behavior of levels that persist for a fixed period of time”, Extremes, 11:2 (2008), 113–133 | DOI | MR | Zbl

[16] M. Ferreira, L. Canto e Castro, “Asymptotic and pre-asymptotic tail behavior of a power max-autoregressive model”, ProbStat Forum, 3 (2010), 91–107 | Zbl

[17] M. Ferreira, L. Canto e Castro, “Modeling rare events through a $p$RARMAX process”, J. Statist. Plann. Inference, 140:11 (2010), 3552–3566 | DOI | MR | Zbl

[18] G. Frahm, “On the extremal dependence coefficient of multivariate distributions”, Statist. Probab. Lett., 76:14 (2006), 1470–1481 | DOI | MR | Zbl

[19] T. Ané, C. Kharoubi, “Dependence structure and risk measure”, J. Business, 76:3 (2003), 411–438

[20] M. Junker, A. May, “Measurement of aggregate risk with copulas”, Econom. J., 8:3 (2005), 428–454 | DOI | MR | Zbl

[21] S. Coles, J. Heffernan, J. A. Tawn, “Dependence measures for extreme value analyses”, Extremes, 2:4 (1999), 339–365 | DOI | Zbl

[22] R. Schmidt, U. Stadtmüller, “Nonparametric estimation of tail dependence”, Scand. J. Statist., 33:2 (2006), 307–335 | DOI | MR | Zbl

[23] J. Klotz, “Statistical inference in Bernoulli trials with dependence”, Ann. Statist., 1:2 (1973), 373–379 | DOI | MR | Zbl