Skip to main content
SpringerLink
Log in

Asymptotic inference for AR(1) processes with (nonnormal) stable errors

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

This is the first of several papers in which we consider problems related to the asymptotic distribution of the least squares estimate of the parameter γ in theAR(1) model

$$X_k = \gamma X_{k - 1} + \varepsilon _k , k = 1,...,n,$$

where εk are independent identically distributed (i.i.d.) random variables in the domain of attraction of a stable law. In §1 we give a summary in the case εk is in the domain of attraction of the normal distribution. In §2 we consider errors in the domain of attraction of a (nonnormal) stable distribution. In §3 we prove a result in the case of the completely asymmetric stable distribution with α=β=1.

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. M. A. Al-Osh and A. A. Alzaid, “First-order integer-valued autoregressive (INAR(1)) process,”J. Time Series Anal.,8, 261–275 (1987).

    MathSciNet  MATH  Google Scholar 

  2. T. W. Anderson, “On asymptotic distributions of estimates of parameters of stochastic difference equations,”Ann. Math. Statist.,30, 676–687 (1959).

    MATH  MathSciNet  Google Scholar 

  3. P. J. Brockwell and R. A. Davis,Times Series Theory and Methods, Springer, New York (1987).

    Google Scholar 

  4. N. H. Chan, “Inference for near-integrated time series with infinite variance,”J.A.S.A.,85, 1069–1074 (1990).

    MATH  Google Scholar 

  5. N. H. Chan, “On the noninvertible moving average time series with infinite variance,”Econ. Theor.,9, 680–685 (1993).

    Google Scholar 

  6. N. H. Chan and L. T. Tran, “On the first-order autoregressive process with infinite variance,”Econ. Theor. 5, 354–362 (1989).

    MathSciNet  Google Scholar 

  7. R. Davis and S. Resnick, “Limit theory for the sample covariance and correlation functions of moving averages,”Ann. Statist.,14, 533–558 (1986).

    MathSciNet  MATH  Google Scholar 

  8. R. G. Laha and V. K. Rohatgi,Probability Theory, Wiley, New York (1979).

    MATH  Google Scholar 

  9. E. McKenzie, “Auto-regressive moving average processes with negative binomial and geometric marginal distributions,”Adv. Appl. Probab.,18, 679–705 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. L. Mijnheer,Sample Path Properties of Stable Processes, Math. Centre, Amsterdam (1975).

    MATH  Google Scholar 

  11. J. L. Mijnheer, “U-statistics and double stable integrals,”Lect. Notes Math.,18, 256–268 (1991).

    MATH  MathSciNet  Google Scholar 

  12. T. v. d. Meer, G. Pap, and M. v. Zuijlen,Asymptotic Inference for Nearly Unstable AR(p) Processes, Tech. Report 9413, Univ. of Nijmegen (1994).

  13. S. Nabeya and B. E. Sørensen, “Asymptotic distributions of the leastsquares estimators and test statistics in the near unit root model with non-zero initial value and local drift and trend,”Econ. Theor.,10, 937–966 (1994).

    Article  Google Scholar 

  14. C. H. Sim, “First-order autoregressive models for gamma and exponential processes,”J. Appl. Probab.,27, 325–332 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Samorodnitsky and J. Szulga, “An asymptotic evaluation of the tail of a multiple symmetric α-stable integral,”Ann. Probab.,17, 1503–1520 (1989).

    MathSciNet  MATH  Google Scholar 

  16. J. S. White, “The limiting distribution of the serial correlation coefficient in the explosive case,”Ann. Math. Statist.,29, 1188–1197 (1958).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Leiden University, P.O. Box 9512, 2300 RA, Leiden, The Netherlands

    J. Mijnheer

Authors
  1. J. Mijnheer
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mijnheer, J. Asymptotic inference for AR(1) processes with (nonnormal) stable errors. J Math Sci 83, 401–406 (1997). https://doi.org/10.1007/BF02400925

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02400925

Keywords

Search

Navigation