TEST

, Volume 22, Issue 4, pp 606–627 | Cite as

Extremes of multivariate ARMAX processes

Original Paper

Abstract

We define a new multivariate time series model by generalizing the ARMAX process in a multivariate way. We give conditions on stationarity and analyze local dependence and domains of attraction. As a consequence of the obtained results, we derive new multivariate extreme value distributions. We characterize the extremal dependence by computing the multivariate extremal index and bivariate upper tail dependence coefficients. An estimation procedure for the multivariate extremal index is presented. We also address the marginal estimation and propose a new estimator for the ARMAX autoregressive parameter.

Keywords

Multivariate extreme value theory Maximum autoregressive processes Multivariate extremal index Tail dependence Asymptotic independence 

Mathematics Subject Classification

60G70 

References

  1. Alpuim MT (1989) An extremal Markovian sequence. J Appl Probab 26:219–232 MathSciNetCrossRefMATHGoogle Scholar
  2. Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New York MATHGoogle Scholar
  3. Canto e Castro L (1992) Sobre a Teoria Assintótica de Extremos. PhD Thesis, FCUL 1992 Google Scholar
  4. Daley D, Haslett J (1982) A thermal energy storage with controlled input. Adv Appl Probab 14(2):257–271 MathSciNetCrossRefMATHGoogle Scholar
  5. Davis R, Resnick S (1989) Basic properties and prediction of max-ARMA processes. Adv Appl Probab 21:781–803 MathSciNetCrossRefMATHGoogle Scholar
  6. Einmahl JHJ, Krajina A, Segers J (2012) An M-estimator for tail dependence in arbitrary dimensions. Ann Stat 40(3):1764–1793 MathSciNetCrossRefMATHGoogle Scholar
  7. Ferreira H (1994) Multivariate extreme values in T-periodic random sequences under mild oscillation restrictions. Stoch Process Appl 49:111–125 CrossRefMATHGoogle Scholar
  8. Ferreira M (2012) Parameter estimation and dependence characterization of the MAR(1) process. ProbStat Forum 5(12):107–111 MathSciNetGoogle Scholar
  9. Ferreira M, Canto e Castro L (2008) Tail and dependence behaviour of levels that persist for a fixed period of time. Extremes 11(2):113–133 MathSciNetCrossRefMATHGoogle Scholar
  10. Ferreira M, Canto e Castro L (2010) Modeling rare events through a pRARMAX process. J Stat Plan Inference 140(11):3552–3566 MathSciNetCrossRefMATHGoogle Scholar
  11. Ferreira H, Ferreira M (2012) On extremal dependence of block vectors. Kybernetika 48(5):988–1006 MathSciNetMATHGoogle Scholar
  12. Greenwood PE, Hooghiemstra G (1988) An extreme-type limit law for a storage process. Math Oper Res 13(2):232–242 MathSciNetCrossRefMATHGoogle Scholar
  13. Haslett J (1979) Problems in the stochastic storage of a solar thermal energy. In: Jacobs O (ed) Analysis and optimization of stochastic systems. Academic Press, London Google Scholar
  14. Heffernan JE, Tawn JA, Zhang Z (2007) Asymptotically (in)dependent multivariate maxima of moving maxima processes. Extremes 10:57–82 MathSciNetCrossRefMATHGoogle Scholar
  15. Huang X (1992) Statistics of bivariate extreme values. Thesis Publishers, Amsterdam Google Scholar
  16. Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London CrossRefMATHGoogle Scholar
  17. Leadbetter MR, Nandagopalan S (1989) On exceedance point processes for stationary sequences under mild oscillation restrictions. In: Hüsler J, Reiss R-D (eds) Extreme value theory. Springer, Berlin, pp 69–80 CrossRefGoogle Scholar
  18. Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, New York CrossRefMATHGoogle Scholar
  19. Lebedev AV (2008) Statistical analysis of first-order MARMA processes. Math Notes 83(4):506–511 MathSciNetCrossRefMATHGoogle Scholar
  20. Ledford A, Tawn JA (1996) Statistics for near independence in multivariate extreme values. Biometrika 83:169–187 MathSciNetCrossRefMATHGoogle Scholar
  21. Ledford A, Tawn JA (1997) Modelling dependence within joint tail regions. J R Stat Soc, Ser B, Stat Methodol 59:475–499 MathSciNetCrossRefMATHGoogle Scholar
  22. Nandagopalan S (1990) Multivariate extremes and estimation of the extremal index. PhD Dissertation, Department of Statistics, University of North, Carolina, USA Google Scholar
  23. Prokhorov YV (1956) Convergence of random processes and limit theorems in probability theory. Theory Probab Appl 1(2):157–214 CrossRefGoogle Scholar
  24. Schmidt R, Stadtmüller U (2006) Nonparametric estimation of tail dependence. Scand J Stat 33:307–335 CrossRefMATHGoogle Scholar
  25. Sibuya M (1960) Bivariate extreme statistics. Ann Inst Stat Math 11:195–210 MathSciNetCrossRefMATHGoogle Scholar
  26. Smith RL, Weissman I (1996) Characterization and estimation of the multivariate extremal index. Technical Report, Univ North, Carolina Google Scholar
  27. Stout W (1974) Almost sure convergence. Academic Press, New York MATHGoogle Scholar
  28. Zhang Z (2006) A new class of tail-dependent time-series models and its applications in financial time series. In: Fomby TB, Terrell D (eds) Econometric analysis of financial and economic time series. Advances in econometrics, vol 20. Emerald Group Publishing Limited, Bingley, pp 317–352 CrossRefGoogle Scholar
  29. Zhang Z (2008) The estimation of M4 processes with geometric moving patterns. Ann Inst Stat Math 60:121–150 CrossRefMATHGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2013

Authors and Affiliations

  1. 1.Center of MathematicsMinho University/DMABragaPortugal
  2. 2.Department of MathematicsUniversity of Beira InteriorCovilhãPortugal

Personalised recommendations