Skip to main content
SpringerLink
Log in

On the Maximum of a Bivariate Max-INAR(1) Process

  • Conference paper
  • First Online:
Recent Developments in Statistics and Data Science (SPE 2021)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 398))

  • 435 Accesses

Abstract

In this paper, we introduce a \(\mathbb {Z}_+^2\)-valued strictly stationary bivariate max-INAR(1) model, which is an extension of the univariate max-INAR(1) model, introduced and studied in [1]. We consider that the marginals have a double geometric distribution in the sense of Marshall and Olkin [2]. As a consequence, we deduce that the innovations have a tail equivalent to a bivariate geometric distribution. By proving that the restriction dependence conditions introduced in [3] hold, we establish asymptotic lower and upper bounds for the distribution function of the double maxima.

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 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
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

References

  1. Scotto, M.G., Weiss, C.H., Möller, T.A., Gouveia, S.: The max-INAR(1) model for count processes. TEST 27, 850–870 (2018). https://doi.org/10.1007/s11749-017-0573-z

    Article  MathSciNet  MATH  Google Scholar 

  2. Marshall, A.W., Olkin, I.: A family of bivariate distributions generated by the bivariate Bernoulli distribution. J. Amer. Stat. Assoc. 80, 332–338 (1985). https://doi.org/10.2307/2287890

    Article  MathSciNet  MATH  Google Scholar 

  3. Hüsler, J.: Multivariate extreme values in stationary random sequences. Stoch. Proc. Appl. 35, 99–108 (1990). https://doi.org/10.1016/0304-4149(90)90125-C

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. McKenzie, E.: Some simple models for discrete variate time series. J. Amer. Water Resour. Assoc. 21, 645–650 (1985). https://doi.org/10.1111/j.1752-1688.1985.tb05379.x

    Article  Google Scholar 

  6. Al-Osh, M.A., Alzaid, A.A.: First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8, 261–275 (1987). https://doi.org/10.1111/j.1467-9892.1987.tb00438.x

    Article  MathSciNet  MATH  Google Scholar 

  7. Pedeli, X., Karlis, D.: A bivariate INAR(1) process with application. Stat. Model. 11(4), 325–349 (2011). https://doi.org/10.1177/1471082X1001100403

    Article  MathSciNet  MATH  Google Scholar 

  8. Silva, I., Silva, M.E., Torres, C.: Inference for bivariate integer-valued moving average models based on binomial thinning operation. J. Appl. Stat. 47, 2546–2564 (2020). https://doi.org/10.1080/02664763.2020.1747411

    Article  MathSciNet  MATH  Google Scholar 

  9. Anderson, C.W.: Extreme value theory for a class of discrete distribution with applications to some stochastic processes. J. Appl. Probab. 7, 99–113 (1970). https://doi.org/10.2307/3212152

    Article  MathSciNet  MATH  Google Scholar 

  10. Dias, S., Temido, M.G.: On the maxima of integer models based on a new thinning operator. In: Oliveira, T., Kitsos, C., Oliveira, A., Grilo, L. (eds.) Recent Studies on Risk Analysis and Statistical Modeling. Contributions to Statistics, pp. 213–226. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76605-8_15

  11. Hall, A.: Extremes of integer-valued moving average models with exponential type tails. Extremes 6, 361–379 (2003). https://doi.org/10.1007/s10687-004-4725-z

    Article  MathSciNet  MATH  Google Scholar 

  12. Hall, A., Scotto, M.G.: Extremes of periodic integer-valued sequences with exponential type tails. REVSTAT 4(3), 249–273 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Hall, A., Temido, M.G.: On the maximum term of MA and Max-AR models with margins in Anderson’s class. Theory Probab. Appl. 51, 291–304 (2007). https://doi.org/10.1137/S0040585X97982347

    Article  MathSciNet  MATH  Google Scholar 

  14. Hall, A., Temido, M.G.: On the max-semistable limit of maxima of stationary sequences with missing values. J. Stat. Plan. Infer. 3, 875–890 (2009). https://doi.org/10.1016/j.jspi.2008.05.038

    Article  MathSciNet  MATH  Google Scholar 

  15. Hüsler, J., Temido, M.G., Valente-Freitas, A.: On the maximum term of a bivariate infinite MA model with integer innovations. Methodol. Comput. Appl. Probab. (2022). https://doi.org/10.1007/s11009-021-09920-3

  16. Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewer for his/her useful comments and suggestions.

The work of the first author was partially supported by Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon—UIDB/04621/2020, funded by the Portuguese Government through FCT/MCTES. The work of the second author was partially supported by the Centre for Mathematics of the University of Coimbra—UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.

Author information

Authors and Affiliations

  1. CEMAT, Department of Mathematics, University of Trás-os-Montes e Alto Douro, Vila Real, Portugal

    Sandra Dias

  2. University of Coimbra, Department of Mathematics, FSTUC, CMUC, Coimbra, Portugal

    Maria da Graça Temido

Authors
  1. Sandra Dias
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria da Graça Temido
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sandra Dias .

Editor information

Editors and Affiliations

  1. NOVA School of Science and Technology, Caparica, Portugal

    Regina Bispo

  2. University of Évora, Évora, Portugal

    Lígia Henriques-Rodrigues

  3. University of Évora, Évora, Portugal

    Russell Alpizar-Jara

  4. University of Edinburgh, Edinburgh, UK

    Miguel de Carvalho

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dias, S., da Graça Temido, M. (2022). On the Maximum of a Bivariate Max-INAR(1) Process. In: Bispo, R., Henriques-Rodrigues, L., Alpizar-Jara, R., de Carvalho, M. (eds) Recent Developments in Statistics and Data Science. SPE 2021. Springer Proceedings in Mathematics & Statistics, vol 398. Springer, Cham. https://doi.org/10.1007/978-3-031-12766-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation