Skip to main content
SpringerLink
Log in

Autoregressive-Type Time Series of Counts

  • Chapter
  • First Online:
Non-Gaussian Autoregressive-Type Time Series

  • 394 Accesses

Abstract

The standard form of an autoregressive model does not help in defining a similar model for discrete time series. A probabilistic operator called binomial thinning is introduced to consolidate the effect of past information to model the future counts. This chapter provides a summary of the autoregressive-type models for generating time series of counts. Even though, the model structures are different, the probabilistic and second-order properties look similar to those of AR models. The models with specified stationary distributions as well as specified innovations are considered. Relevant estimation problems are also studied.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Reference [12] defined INAR(1) model assuming that \(\{Z_t\}\) is a sequence of uncorrelated non-negative integer-valued rvs. We will continue with iid sequence to retain the Markov property of \(\{X_t\}.\)

  2. 2.

    See Chap. 6 for more details on the autoregressive models with stable innovations.

  3. 3.

    Structure of this innovation distribution is similar to that of EAR(1) model discussed in Chap. 3.

  4. 4.

    A summary of the relevant estimation methods and the related results are provided in Chapter 2 of this book. Readers may refer to that chapter for details.

  5. 5.

    These regularity conditions and the relevant results from [37] are listed in Chap. 2 for an easy reference.

  6. 6.

    Some of the basic properties of Godambe’s optimal estimating functions (EF) and the related asymptotic results are summarized in Chap. 2. In particular, we use Theorem in the following discussion.

  7. 7.

    The CAN property follows from Theorem 2.7.3. See also [43].

References

  1. C. Weiss, An Introduction to Discrete-Valued Time Series, 1st edn. (England, Wiley, Chichester, 2018)

    Google Scholar 

  2. M.B. Rajarshi, Statistical Inference for Discrete Time Stochastic Processes (Springer, 2012)

    Google Scholar 

  3. P.A. Jacobs, Cyclic queuing network with dependent exponential service time. J. Appl. Probab. 15, 573–589 (1978)

    Article  MathSciNet  Google Scholar 

  4. P.A. Jacobs, P.A.W. Lewis, Discrete time series generated by mixtures i. Correlational and runs properties. J. R. Stat. Soc. B. 40, 94–105 (1978)

    Google Scholar 

  5. P.A. Jacobs, P.A.W. Lewis, Discrete time series generated by mixtures i: Asymptotic properties. J. R. Stat. Soc. B. 40, 222–228 (1978)

    Google Scholar 

  6. P.A. Jacobs, P.A.W. Lewis, Stationary discrete autoregressive-moving average time series generated by mixtures. J. Time Ser. Anal. 4, 19–36 (1983)

    Google Scholar 

  7. E. Mckenzie, Some simple models for discrete variate time series. Water Resour. Bull. 21(4), 645–650 (1985)

    Article  Google Scholar 

  8. M.G. Scotto, C.H. Weiß, S. Gouveia, On mixed AR (1) time series model with approximated beta marginal. Stat. Model. 15(6), 590–618 (2015)

    Google Scholar 

  9. E. Mckenzie, Autoregressive-moving average processes with negative-binomial and geometric marginal distributions. Adv. Appl. Probab. 18, 679–705 (1986)

    Article  MathSciNet  Google Scholar 

  10. E. Mckenzie, Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Probab. 20, 822–835 (1988)

    Google Scholar 

  11. F.W. Steutel, K. Van Harn, Discrete analogues of self-decomposability and stability. Ann. Probab. 7, 893–899 (1979)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  13. A. Alzaid, M. Al-Osh, An integer-valued pth-order autoregressive structure (INAR(p)) process. J. Appl. Probab. 27, 314–324 (1990)

    Google Scholar 

  14. J.-G. Du, Y. Li, The integer-valued autoregressive (INAR(p)) model. J. Time Ser. Anal. 12, 129–142 (1991)

    Google Scholar 

  15. A. Latour, Existence and stochastic structure of a non-negative integer-valued autoregressive process. J. Time Ser. Anal. 19(4), 439–455 (1998)

    Article  MathSciNet  Google Scholar 

  16. R. Bu, B. McCabe, K. Hadri, Maximum likelihood estimation of higher-order integer-valued autoregressive processes. J. Time Ser. Anal. 29(6), 973–994 (2008)

    Article  MathSciNet  Google Scholar 

  17. M.M. Ristic, H.S. Bakouch, A.S. Nasti’c, A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. J. Stat. Plann. Inference 139(7), 2218–2226 (2009)

    Google Scholar 

  18. E. Mckenzie, Innovation distributions for gamma and negative binomial autoregressions. Scand. J. Stat. 14, 79–85 (1987)

    MathSciNet  MATH  Google Scholar 

  19. M.M. Ristic, A.S. Nasti’c, H.S. Bakouch, Estimation in an integer-valued autoregressive process with negative binomial marginals (NBINAR(1)). Commun. Stat. Theory Methods 41(4), 606–618 (2012)

    Google Scholar 

  20. M.A. Al-Osh, A.A. Alzaid, Binomial autoregressive moving average models. Commun. Stat. Stoch. Models 7, 261–282 (1991)

    Google Scholar 

  21. X. Pedeli, A.C. Davison, K. Fokianos, Likelihood estimation for the INAR(p) model by saddlepoint approximation. J. Am. Stat. Assoc. 110(511), 1229–1238 (2015)

    Google Scholar 

  22. A. Noack, A class of random variables with discrete distributions. Ann. Math. Stat. 21, 127–132 (1950)

    Article  MathSciNet  Google Scholar 

  23. M. Bourguignon, K.L.P. Vasconcellos, First order non-negative integer valued autoregressive processes with power series innovations. Braz. J. Prob. Stat. 29(1), 71–93 (2015)

    Article  MathSciNet  Google Scholar 

  24. J. Huang, F. Zhu, A new first-order integer-valued autoregressive model with bell innovations. Entropy 23(1), 1–17 (2021)

    Article  MathSciNet  Google Scholar 

  25. F. Castellares, S.L.P. Ferrari, A.J. Lemonte, On the bell distribution and its associated regression model for count data. Appl. Math. Model. 56, 172–185 (2018)

    Article  MathSciNet  Google Scholar 

  26. S.L. Zeger, A regression model for time series of counts. Biometrika 75, 621–629 (1988)

    Article  MathSciNet  Google Scholar 

  27. S.L. Zeger, B. Qaqish, Markov regression models for time series: A quasi-likelihood approach. Biometrics 44, 1019–1031 (1988)

    Google Scholar 

  28. D.R. Cox, Statistical analysis of time series: some recent developments (with discussion). Scandinvian J. Stat. 8, 93–115 (1981)

    MATH  Google Scholar 

  29. A.C. Cameron, P.K. Trivedi, Regression Analysis of Count Data, 2nd edn. (Cambridge University Press, New York., 2013)

    Google Scholar 

  30. R.A. Davis, K. Fokianos, S.H. Holan, H. Joe, J. Livsey, R. Lund, V. Pipiras, N. Ravishanker, Count time series: A methodological review. J. Am. Stat. Assoc. (2021). https://doi.org/10.1080/01621459.2021.1904957

  31. K. Fokianos, Count time series models, in Handbook of Statistics: Time Series Analysis–Methods and Applications, ed. by T.S. Rao, S.S. Rao, C.R. Rao, vol. 30 (2012), pp. 315–347

    Google Scholar 

  32. K. Fokianos, A. Rahbek, D. Tjøstheim, Poisson autoregression. J. Am. Stat. Assoc. 1004, 1430–1439 (2009)

    Article  MathSciNet  Google Scholar 

  33. F. Zhu, A negative binomial integer-valued garch model. J. Time Ser. Anal. 32(1), 54–67 (2011)

    Article  MathSciNet  Google Scholar 

  34. P.J. Brockwell, R.A. Davis, Time Series Theory and Methods, 2nd edn. (Springer, New York., 2006)

    Google Scholar 

  35. I. Silva, M.E. Silva, Asymptotic distribution of the yule-walker estimator for INAR(p) processes. Stat. Probab. Lett. 76, 1655–1663 (2006)

    Google Scholar 

  36. C.W. Park, Y. Oh, Some asymptotic properties in INAR(1) processes with Poisson marginals. tStat. Pap. 38, 287–302 (1997)

    Google Scholar 

  37. L.A. Klimko, P.I. Nelson, On conditional least squares estimation for stochastic processes. Ann. Stat. 629–642 (1978)

    Google Scholar 

  38. B.P.M. Freeland, R.K. McCabe. Asymptotic properties of CLS estimators in the Poisson AR(1) model. Stat. Probab. Lett. 73(2), 147–153 (2004)

    Google Scholar 

  39. P. Billingsley, Statistical Inference for Markov Processes, vol. 2 (University of Chicago Press, 1961)

    Google Scholar 

  40. B.P.M. Freeland, R.K. McCabe, Analysis of low count time series data by Poisson autoregression. J. Time Ser. Anal. 25(5), 701–722 (2004)

    Google Scholar 

  41. H.E. Daniels, Saddlepoint approximations in statistics. Ann. Math. Stat. 25, 631–650 (1954)

    Article  MathSciNet  Google Scholar 

  42. J.E. Kolassa, Series Approximation Methods in Statistics, 3rd edn. (Springer, New York., 2006)

    Google Scholar 

  43. S.Y. Hwang, I.V. Basawa, Godambe estimating functions and asymptotic optimal inference. Stat. Probab. Lett. 81(8), 1121–1127 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Cochin University of Science and Technology, Kochi, Kerala, India

    N. Balakrishna

Authors
  1. N. Balakrishna
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Balakrishna .

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Balakrishna, N. (2021). Autoregressive-Type Time Series of Counts. In: Non-Gaussian Autoregressive-Type Time Series. Springer, Singapore. https://doi.org/10.1007/978-981-16-8162-2_7

Download citation

Publish with us

Policies and ethics

Search

Navigation