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ArticleOriginal scientific text

Title

Extremal (in)dependence of a maximum autoregressive process

Authors 1

Affiliations

  1. Center of Mathematics of Minho University, Campus de Gualtar, 4710 - 057 Braga, Portugal

Abstract

Maximum autoregressive processes like MARMA (Davis and Resnick, [5] 1989) or power MARMA (Ferreira and Canto e Castro, [12] 2008) have singular joint distributions, an unrealistic feature in most applications. To overcome this pitfall, absolute continuous versions were presented in Alpuim and Athayde [2] (1990) and Ferreira and Canto e Castro [14] (2010b), respectively. We consider an extended version of absolute continuous maximum autoregressive processes that accommodates both asymptotic tail dependence and independence. A full characterization of the bivariate lag-m tail dependence is presented. This will be useful in an adjustment procedure of the model to real data. An illustration with financial data is presented at the end.

Keywords

ENG
extreme value theory, autoregressive processes, tail dependence, asymptotic tail independence

Bibliography

  1. M.T. Alpuim, An extremal Markovian sequence, J. Appl. Probab. 26 (1989) 219-232. doi: 10.2307/3214030.
  2. M.T. Alpuim and E. Athayde, On the stationary distribution of some extremal Markovian sequences, J. Appl. Probab. 27 (1990) 291-302. doi: 10.2307/3214648.
  3. J. Beirlant, Y. Goegebeur, J. Segers and J. Teugels, Statistics of Extremes: Theory and Application (John Wiley, Chichester, UK, 2004). doi: 10.1002/0470012382.
  4. P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer-Verlag, Berlin, 1997).
  5. R. Davis and S. Resnick, Basic properties and prediction of max-ARMA processes, Adv. Appl. Probab. 21 (1989) 781-803. doi: 10.2307/1427767.
  6. A.L.M. Dekkers, J.H.J. Einmahl and 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.
  7. D. Dietrich, L. de Haan and J. Hüsler, Testing extreme value conditions, Extremes 5 (2000) 71-85. doi: 10.1023/A:1020934126695.
  8. G. Draisma, H. Drees, A. Ferreira and L. de Haan, Bivariate tail estimation: dependence in asymptotic independence, Bernoulli 10 (2004) 251-280. doi: 10.3150/bj/1082380219.
  9. H. Drees, Extreme quantile estimation for dependent data with applications to finance, Bernoulli 9 (2003) 617-657. doi: 10.3150/bj/1066223272.
  10. M. Ferreira, On tail dependence: a characterization for first-order max-autoregressive processes, Math. Notes 90 (2011) 103-114. doi: 10.1134/S0001434611110277.
  11. M. Ferreira, Parameter estimation and dependence characterization of the MAR(1) process, ProbStat Forum 5 (2012) 107-111.
  12. M. Ferreira and L. Canto e Castro, Tail and dependence behaviour of levels that persist for a fixed period of time, Extremes 11 (2008) 113-133. doi: 10.1007/s10687-007-0046-y.
  13. M. Ferreira and L. Canto e Castro, Asymptotic and pre-asymptotic tail behavior of a power max-autoregressive model, ProbStat Forum 3 (2010a) 91-107.
  14. M. Ferreira and L. Canto e Castro, Modeling rare events through a pRARMAX process, J. Statist. Plann. Inference 140 (2010b) 3552-3566. doi: 10.1016/j.jspi.2010.05.024.
  15. M. Ferreira and L. Canto e Castro, A Method For Fitting A pRARMAX Model: An Application To Financial Data, in: Proceedings of The World Congress on Engineering 2010 Vol. III, Lecture Notes in Engineering and Computer Science (Ed(s)), (London, U.K., 2010c) 2022-2026.
  16. G. Frahm, M. Junker and R. Schmidt, Estimating the tail-dependence coefficient: properties and pitfalls, Insurance Math. Econom. 37 (2005) 80-100. doi: 10.1016/j.insmatheco.2005.05.008.
  17. J.E. Heffernan, J.A. Tawn and Z. Zhang, Asymptotically (in)dependent multivariate maxima of moving maxima processes, Extremes 10 (2007) 57-82. doi: 10.1007/s10687-007-0035-1.
  18. 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.
  19. A.V. Lebedev, Statistical analysis of first-order MARMA processes, Math. Notes 83 (2008) 506-511. doi: 10.1134/S0001434608030243.
  20. Z. Zhang, A New Class of Tail-Dependent Time Series Models and Its Applications in Financial Time Series, Advances in Econometrics 20B (2005) 323-358.
Discussiones Mathematicae Probability and Statistics
2013/33/1-2
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Pages:
47-64
Main language of publication
English
Received
2013-03-04
Accepted
2013-09-16
Published
2013

DOI: 10.7151/dmps1150

Exact and natural sciences