Skip to main content
SpringerLink
Log in

Statistical analysis of first-order MARMA processes

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In this paper, we consider processes of the MARMA type that can be derived from the classical ARMA processes by replacing summation by the maximum operation. It is assumed that the innovations and the values of the process have the standard Fréchet distribution. For simple MARMA processes of first order, certain numerical characteristics are calculated. Sign tests and rank statistical methods for parameter estimation are developed. The characterization relations that can be used for the identification of models are justified.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance, in Applications of Mathematics (Springer-Verlag, Berlin, 1997), Vol. 33.

    Google Scholar 

  2. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, Oxford, 1982; Inst. of Comp. Studies, Moscow, 2002).

    MATH  Google Scholar 

  3. A. J. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management, in Princeton Series in Finance: Concepts, Techniques, and Tools (Princeton Univ. Press, Princeton, NJ, 2005).

    Google Scholar 

  4. R. A. Davis and S. I. Resnick, “Basic properties and prediction of max-ARMA processes,” Adv. in Appl. Probab. 21(4), 781–803 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  5. R. A. Davis and S. I. Resnick, “Prediction of stationary max-stable processes,” Ann. Appl. Probab. 3(2), 497–525 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  6. I. Helland and T. Nilsen, “On a general random exchange model,” J. Appl. Probab. 13(4), 781–790 (1976).

    Article  MathSciNet  Google Scholar 

  7. D. Daley and J. Haslet, “A thermal energy storage with controlled input,” Adv. in Appl. Probab. 14(2), 257–271 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  8. S. G. Coles, “Regional modelling of extreme storms via max-stable processes,” J. Roy. Statist. Soc. Ser. B 55(4), 797–816 (1993).

    MATH  MathSciNet  Google Scholar 

  9. Z. Zhang and R. L. Smith, Modelling Financial Time Series Data as Moving Maxima Processes, Tech. Rep. Dept. Stat. (Univ. North Carolina, Chapel Hill, NC, 2001); http://www.stat.unc.edu/faculty/rs/papers/RLS_Papers.html.

    Google Scholar 

  10. M. T. Alpuim, “An extremal Markovian sequence,” J. Appl. Probab. 26(2), 219–232 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  11. M. T. Alpuim, N. A. Catkan, and J. Hüsler, “Extremes and clustering of nonstationary max-AR(1) sequences,” Stochastic Process. Appl 56(1), 171–184 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Box and G. Jenkins, Time Series Analysis: Forecasting and Control (McGraw-Hill, Holden-Day, San Francisco, 1970; Mir, Moscow, 1974).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    A. V. Lebedev

Authors
  1. A. V. Lebedev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Lebedev.

Additional information

Original Russian Text © A. V. Lebedev, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 4, pp. 552–558.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lebedev, A.V. Statistical analysis of first-order MARMA processes. Math Notes 83, 506–511 (2008). https://doi.org/10.1134/S0001434608030243

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434608030243

Key words

Search

Navigation