Skip to main content
SpringerLink
Log in

New Discrete Bilal Distribution and Associated INAR(1) Process

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

The significance of applications in count data modeling to real-life scenarios has been emphasized in numerous studies. The present paper focuses on a discrete version of the Bilal distribution, the new discrete Bilal distribution and its structural properties. The proposed distribution’s mathematical and statistical characteristics are thoroughly investigated. The model parameter estimation is carried out by maximum likelihood estimation, and a simulation study is performed. The performance of the proposed model is evaluated by real data analysis. The significance of the proposed distribution in time series is demonstrated with the associated INAR(1) process having innovations as the proposed model. The parameter estimation of the INAR(1) process is carried out using conditional maximum likelihood, conditional least squares and Yule–Walker estimation procedures, and real data is used to determine the empirical importance of the proposed process.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

REFERENCES

  1. A. Abd-Elrahman, ‘‘Utilizing ordered statistics in lifetime distributions production: A new lifetime distribution and applications,’’ J. Prob. Stat. Sci. 11, 153–164 (2013).

    MathSciNet  Google Scholar 

  2. A. Abd-Elrahman, ‘‘A new two-parameter lifetime distribution with decreasing, increasing or upside-down bathtub-shaped failure rate,’’ Commun. Stat. Theory Methods 46, 8865–8880 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Altun, M. El-Morshedy, and M. Eliwa, ‘‘A new regression model for bounded response variable: An alternative to the beta and unit-Lindley regression models,’’ Plos One 16 (2021).

  4. E. Altun, ‘‘A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models,’’ Math. Slov. 70, 979–994 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Maya, M. Irshad, and S. Arun, ‘‘Application of U-statistics in estimation of scale parameter of bilal distribution,’’ Philippine Stat. 70, 67–82 (2021).

    Google Scholar 

  6. R. Maya, M. Irshad, and S. Arun, ‘‘Farlie–Gumbel–Morgenstern bivariate Bilal distribution and its inferential aspects using concomitants of order statistics,’’ J. Prob. Stat. Sci. 19, 1–20 (2021).

    Google Scholar 

  7. C. Weiß, An Introduction to Discrete-Valued Time Series (Wiley, New York, 2018).

    Book  MATH  Google Scholar 

  8. M. Huang and K. Fung, ‘‘Intervened truncated Poisson distribution,’’ Sankhy\(\bar{a}\): Ind. J. Stat., Ser. B 51, 302–310 (1989).

    MathSciNet  MATH  Google Scholar 

  9. A. Eldeeb, M. Ahsan-Ul-Haq, and A. Babar, ‘‘A discrete analog of inverted Topp–Leone distribution: Properties, estimation and applications,’’ Int. J. Anal. Appl. 19, 695–708 (2021).

    Google Scholar 

  10. T. Hussain and M. Ahmad, ‘‘Discrete inverse Rayleigh distribution,’’ Pakist. J. Stat. 30 (2014).

  11. E. Gomez-Deniz and E. Calderin-Ojeda, ‘‘The discrete Lindley distribution: Properties and applications,’’ J. Stat. Comput. Simul. 81, 1405–1416 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Al-Babtain, A. Ahmed, and A. Afify, ‘‘A new discrete analog of the continuous Lindley distribution with reliability applications,’’ Entropy 22, 603 (2020).

    Article  MathSciNet  Google Scholar 

  13. A. Eldeeb, M. Haq, M. Eliwa, and Q. Cell, ‘‘A discrete Ramos–Louzada distribution for asymmetric and over-dispersed data with leptokurtic-shaped: Properties and various estimation techniques with inference,’’ AIMS Math. 7, 1726–1741 (2022).

    Article  MathSciNet  Google Scholar 

  14. A. Buddana and T. Kozubowski, ‘‘Discrete Pareto distributions,’’ Econ. Qual. Control 29, 143–156 (2014).

    Article  MATH  Google Scholar 

  15. M. El-Morshedy, M. Eliwa, and E. Altun, ‘‘Discrete Burr–Hatke distribution with properties, estimation methods and regression model,’’ IEEE Access 8, 74359–74370 (2020).

    Article  Google Scholar 

  16. E. McKenzie, ‘‘Some simple models for discrete variate time series,’’ J. Am. Water Resour. Assoc. 21, 645–650 (1985).

    Article  Google Scholar 

  17. M. Aghababaei Jazi, G. Jones, and C. Lai, ‘‘Integer valued AR (1) with geometric innovations,’’ J. Iran. Stat. Soc. 11, 173–190 (2012).

    MathSciNet  MATH  Google Scholar 

  18. E. Altun, ‘‘A new generalization of geometric distribution with properties and applications,’’ Commun. Stat.-Simul. Comput. 49, 793–807 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Schweer and C. Weiß, ‘‘Compound Poisson INAR (1) processes: Stochastic properties and testing for overdispersion,’’ Comput. Stat. Data Anal. 77, 267–284 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Jazi, G. Jones, and C. Lai, ‘‘First-order integer valued AR processes with zero-inflated Poisson innovations,’’ J. Time Ser. Anal. 33, 954–963 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Kemp, ‘‘Characterizations of a discrete normal distribution,’’ J. Stat. Plann. Inference 63, 223–229 (1997).

    Article  MathSciNet  MATH  Google Scholar 

Download references

ACKNOWLEDGMENTS

The authors thank the referees for the valuable suggestions and constructive comments which improved the paper.

Author information

Authors and Affiliations

  1. College of Statistical and Actuarial Sciences, University of the Punjab, Lahore, Pakistan

    M. Ahsan-ul-Haq

  2. Department of Statistics, Cochin University of Science and Technology, 682 022, Cochin, Kerala, India

    M. R. Irshad & E. S. Muhammed Ahammed

  3. Department of Statistics, University College, 695034, Thiruvananthapuram, Kerala, India

    R. Maya

Authors
  1. M. Ahsan-ul-Haq
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. R. Irshad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. S. Muhammed Ahammed
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Maya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to M. Ahsan-ul-Haq, M. R. Irshad, E. S. Muhammed Ahammed or R. Maya.

Additional information

(Submitted by A. I. Volodin)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahsan-ul-Haq, M., Irshad, M.R., Muhammed Ahammed, E.S. et al. New Discrete Bilal Distribution and Associated INAR(1) Process. Lobachevskii J Math 44, 3647–3662 (2023). https://doi.org/10.1134/S1995080223090020

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080223090020

Keywords:

Search

Navigation