PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2013 | 33 | 1-2 | 65-77
Tytuł artykułu

On the tail index estimation of an autoregressive Pareto process

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper we consider an autoregressive Pareto process which can be used as an alternative to heavy tailed MARMA. We focus on the tail behavior and prove that the tail empirical quantile function can be approximated by a Gaussian process. This result allows to derive a class of consistent and asymptotically normal estimators for the shape parameter. We will see through simulation that the usual estimation procedure based on an i.i.d. setting may fall short of the desired precision.
Twórcy
  • Center of Mathematics of Minho University, Campus de Gualtar, 4710 - 057 Braga, Portugal
Bibliografia
  • [1] B.C. Arnold, Pareto Distributions (International Cooperative Publishing House, Fairland, 1983).
  • [2] B.C. Arnold, Pareto processes, in: Handbook of Statistics, D.N. Shanbhag and C.R. Rao, eds., Vol. 19, (Elsevier Science B.V., 2001).
  • [3] R. Davis and S. Resnick, Basic properties and prediction of max-ARMA processes, Adv. Appl. Probab. 21 (1989) 781-803. doi: 10.1214/aos/1176347397.
  • [4] 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.
  • [5] H. Drees, On smooth statistical tail functionals, Scand. J. Statist. 25 (1998a) 187-210. doi: 10.1111/1467-9469.00097.
  • [6] H. Drees, A general class of estimators of the extreme value index, J. Statist. Plann. Inference 66) 1998b (95-112. doi: 10.1016/S0378-3758(97)00076-1.
  • [7] H. Drees, Extreme quantile estimation for dependent data with applications to finance, Bernoulli 9 (2003) 617-657. doi: 0.3150/bj/1066223272.
  • [8] M. Ferreira, On the extremal behavior of a pareto process: an alternative for armax modeling, Kybernetika 48 (2012) 31-49.
  • [9] M. Ferreira, Tail dependence of a Pareto process, accepted for publication in Studies in Theoretical and Applied Statistics - Selected Papers of the Statistical Societies, Springer.
  • [10] H. Ferreira and M. Ferreira, Tail dependence between order statistics, J. Multivariate Anal. 105 (2012) 176-192. doi: 10.1016/j.jmva.2011.09.001.
  • [11] H. Ferreira and M. Ferreira, Fragility Index of block tailed vectors, J. Statist. Plann. Inference 142 (2012) 1837-1848. doi: 10.1016/j.jspi.2012.01.021.
  • [12] J.L. Geluk, L. De Haan and C.G. De Vries, Weak and strong financial fragility. Tinbergen Institute Discussion Paper, TI 2007-023/2, 2007.
  • [13] 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.
  • [14] J.R.M. Hosking and J.R. Wallis, Parameter and quantile estimation for the generalized Pareto distribution, Technometrics 29 (1987) 339-349. doi: 10.1080/00401706.1987.10488243.
  • [15] T. Hsing, On tail estimation using dependent data, Ann. Statist. 19 (1991) 1574-1569. doi: 10.1214/aos/1176348261.
  • [16] H. Joe, Multivariate Models and Dependence Concepts (Chapman & Hall, London, 1997). doi: 10.1201/b13150.
  • [17] S.D. Krishnarani and K. Jayakumar, A class of autoregressive processes, Statist. Probab. Lett. 78 (2008) 1355-1361. doi: 10.1016/j.spl.2007.12.019.
  • [18] A.V. Lebedev, Statistical analysis of first-order MARMA processes, Mathematical Notes 83 (2008) 506-511. doi: 10.1134/S0001434608030243.
  • [19] J. Pickands III, Statistical inference using extreme order statistics, Ann. Statist. 3 (1975) 119-131. doi: 10.1214/aos/1176343003.
  • [20] S. Resnick and C. Stărică, Consistency of Hill's estimator for dependent data, J. Appl. Probab. 32 (1995) 139-167. doi: 10.2307/3214926.
  • [21] S. Resnick and C. Stărică, Tail index estimation for dependent data, Ann. Appl. Probab. 8 (1998) 1156-1183. doi: 10.1214/aoap/1028903376 .
  • [22] H. Rootzén, M.R. Leadbetter and L. de Haan, Tail and Quantile Estimation for Strongly Mixing Stationary Sequences. Technical Report, UNC Center for Stochastic Processes, 1990.
  • [23] M. Sibuya, Bivariate extreme statistics, Ann. Inst. Statist. Math. 11 (1960) 195-210. doi: 10.1007/BF01682329.
  • [24] R.L. Smith, Estimating tails of probability distributions, Ann. Statist. 15 (1987) 1174-1207. doi: 10.1214/aos/1176350499.
  • [25] H.C. Yeh, B.C. Arnold and C.A. Robertson, Pareto processes, J. Appl. Probab. 25 (1988) 291-301. doi: 10.2307/3214437.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1149
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.