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2013 | 33 | 1-2 | 47-64
Tytuł artykułu

Extremal (in)dependence of a maximum autoregressive process

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Języki publikacji
EN
Abstrakty
EN
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.
Twórcy
  • Center of Mathematics of Minho University, Campus de Gualtar, 4710 - 057 Braga, Portugal
Bibliografia
  • [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.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1150
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