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Extremes of multivariate ARMAX processes

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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.

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References

  • Alpuim MT (1989) An extremal Markovian sequence. J Appl Probab 26:219–232

    Article  MathSciNet  MATH  Google Scholar 

  • Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New York

    MATH  Google Scholar 

  • Canto e Castro L (1992) Sobre a Teoria Assintótica de Extremos. PhD Thesis, FCUL 1992

  • Daley D, Haslett J (1982) A thermal energy storage with controlled input. Adv Appl Probab 14(2):257–271

    Article  MathSciNet  MATH  Google Scholar 

  • Davis R, Resnick S (1989) Basic properties and prediction of max-ARMA processes. Adv Appl Probab 21:781–803

    Article  MathSciNet  MATH  Google Scholar 

  • Einmahl JHJ, Krajina A, Segers J (2012) An M-estimator for tail dependence in arbitrary dimensions. Ann Stat 40(3):1764–1793

    Article  MathSciNet  MATH  Google Scholar 

  • Ferreira H (1994) Multivariate extreme values in T-periodic random sequences under mild oscillation restrictions. Stoch Process Appl 49:111–125

    Article  MATH  Google Scholar 

  • Ferreira M (2012) Parameter estimation and dependence characterization of the MAR(1) process. ProbStat Forum 5(12):107–111

    MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Ferreira M, Canto e Castro L (2010) Modeling rare events through a pRARMAX process. J Stat Plan Inference 140(11):3552–3566

    Article  MathSciNet  MATH  Google Scholar 

  • Ferreira H, Ferreira M (2012) On extremal dependence of block vectors. Kybernetika 48(5):988–1006

    MathSciNet  MATH  Google Scholar 

  • Greenwood PE, Hooghiemstra G (1988) An extreme-type limit law for a storage process. Math Oper Res 13(2):232–242

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Heffernan JE, Tawn JA, Zhang Z (2007) Asymptotically (in)dependent multivariate maxima of moving maxima processes. Extremes 10:57–82

    Article  MathSciNet  MATH  Google Scholar 

  • Huang X (1992) Statistics of bivariate extreme values. Thesis Publishers, Amsterdam

    Google Scholar 

  • Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London

    Book  MATH  Google Scholar 

  • 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

    Chapter  Google Scholar 

  • Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, New York

    Book  MATH  Google Scholar 

  • Lebedev AV (2008) Statistical analysis of first-order MARMA processes. Math Notes 83(4):506–511

    Article  MathSciNet  MATH  Google Scholar 

  • Ledford A, Tawn JA (1996) Statistics for near independence in multivariate extreme values. Biometrika 83:169–187

    Article  MathSciNet  MATH  Google Scholar 

  • Ledford A, Tawn JA (1997) Modelling dependence within joint tail regions. J R Stat Soc, Ser B, Stat Methodol 59:475–499

    Article  MathSciNet  MATH  Google Scholar 

  • Nandagopalan S (1990) Multivariate extremes and estimation of the extremal index. PhD Dissertation, Department of Statistics, University of North, Carolina, USA

  • Prokhorov YV (1956) Convergence of random processes and limit theorems in probability theory. Theory Probab Appl 1(2):157–214

    Article  Google Scholar 

  • Schmidt R, Stadtmüller U (2006) Nonparametric estimation of tail dependence. Scand J Stat 33:307–335

    Article  MATH  Google Scholar 

  • Sibuya M (1960) Bivariate extreme statistics. Ann Inst Stat Math 11:195–210

    Article  MathSciNet  MATH  Google Scholar 

  • Smith RL, Weissman I (1996) Characterization and estimation of the multivariate extremal index. Technical Report, Univ North, Carolina

  • Stout W (1974) Almost sure convergence. Academic Press, New York

    MATH  Google Scholar 

  • 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

    Chapter  Google Scholar 

  • Zhang Z (2008) The estimation of M4 processes with geometric moving patterns. Ann Inst Stat Math 60:121–150

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We are very grateful to the referees for their valuable corrections and suggestions. Helena Ferreira was partially supported by the research unit “Centro de Matemática” of the University of Beira Interior and the research project PTDC/MAT/108575/2008 through the Foundation for Science and Technology (FCT) co-financed by FEDER/COMPETE. Marta Ferreira was financed by FEDER Funds through “Programa Operacional Factores de Competitividade—COMPETE” and by Portuguese Funds through FCT—“Fundação para a Ciência e a Tecnologia”, within the Project Est-C/MAT/UI0013/2011.

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Authors and Affiliations

  1. Center of Mathematics, Minho University/DMA, Braga, Portugal

    Marta Ferreira

  2. Department of Mathematics, University of Beira Interior, Covilhã, Portugal

    Helena Ferreira

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  1. Marta Ferreira
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  2. Helena Ferreira
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Correspondence to Marta Ferreira.

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Ferreira, M., Ferreira, H. Extremes of multivariate ARMAX processes. TEST 22, 606–627 (2013). https://doi.org/10.1007/s11749-013-0326-6

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