IDEAS home Printed from https://ideas.repec.org/a/spr/compst/v40y2025i2d10.1007_s00180-024-01511-3.html

Monitoring mean of INAR(1) process with discrete mixture exponential innovations

Author

Listed:
  • M. R. Irshad

    (Cochin University of Science and Technology)

  • Muhammed Ahammed

    (Cochin University of Science and Technology)

  • R. Maya

    (University College)

Abstract

This paper presents a discrete counterpart of the mixture exponential distribution, namely discrete mixture exponential distribution, by utilizing the survival discretization method. The moment generating function and associated moment measures are discussed. The distribution’s hazard rate function can assume increasing or decreasing forms, making it adaptable for diverse fields requiring count data modeling. The paper delves into two parameter estimation methods and evaluates their performance through a Monte Carlo simulation study. The applicability of this distribution extends to time series analysis, particularly within the framework of the first-order integer-valued autoregressive process. Consequently, an INAR(1) process with discrete mixture exponential innovations is proposed, outlining its fundamental properties, and the performance of conditional maximum likelihood and conditional least squares estimation methods is evaluated through a simulation study. Real data analysis showcases the proposed model’s superior performance compared to alternative models. Additionally, the paper explores quality control applications, addressing serial dependence challenges in count data encountered in production and market management. As a result, the INAR(1)DME process is employed to explore control charts for monitoring autocorrelated count data. The performance of two distinct control charts, the cumulative sum chart and the exponentially weighted moving average chart, are evaluated for their effectiveness in detecting shifts in the process mean across various designs. A bivariate Markov chain approach is used to estimate the average run length and their deviations for these charts, providing valuable insights for practical implementation. The nature of design parameters to improve the robustness of process monitoring under the considered charts is examined through a simulation study. The practical superiority of the proposed charts is demonstrated through effective modeling with real data, surpassing competing models.

Suggested Citation

  • M. R. Irshad & Muhammed Ahammed & R. Maya, 2025. "Monitoring mean of INAR(1) process with discrete mixture exponential innovations," Computational Statistics, Springer, vol. 40(2), pages 821-862, February.
    • Handle: RePEc:spr:compst:v:40:y:2025:i:2:d:10.1007_s00180-024-01511-3
    DOI: 10.1007/s00180-024-01511-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00180-024-01511-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00180-024-01511-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. A. Alzaid & M. Al‐Osh, 1988. "First‐Order Integer‐Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 42(1), pages 53-61, March.
    2. Christian Weiß & Murat Testik, 2011. "The Poisson INAR(1) CUSUM chart under overdispersion and estimation error," IISE Transactions, Taylor & Francis Journals, vol. 43(11), pages 805-818.
    3. Harvey, Andrew C & Fernandes, C, 1989. "Time Series Models for Count or Qualitative Observations," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(4), pages 407-417, October.
    4. Cong Li & Haixiang Zhang & Dehui Wang, 2022. "Modelling and monitoring of INAR(1) process with geometrically inflated Poisson innovations," Journal of Applied Statistics, Taylor & Francis Journals, vol. 49(7), pages 1821-1847, May.
    5. Harvey, Andrew C & Fernandes, C, 1989. "Time Series Models for Count or Qualitative Observations: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(4), pages 422-422, October.
    6. Mansour Aghababaei Jazi & Geoff Jones & Chin-Diew Lai, 2012. "First-order integer valued AR processes with zero inflated poisson innovations," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(6), pages 954-963, November.
    7. Emrah Altun & Naushad Mamode Khan, 2022. "Modelling with the Novel INAR(1)-PTE Process," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1735-1751, September.
    8. Radhakumari Maya & Christophe Chesneau & Anuresha Krishna & Muhammed Rasheed Irshad, 2022. "Poisson Extended Exponential Distribution with Associated INAR(1) Process and Applications," Stats, MDPI, vol. 5(3), pages 1-18, August.
    9. A. Alzaid & M. Al-Osh, 1993. "Some autoregressive moving average processes with generalized Poisson marginal distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(2), pages 223-232, June.
    10. Cong Li & Dehui Wang & Fukang Zhu, 2019. "Detecting mean increases in zero truncated INAR(1) processes," International Journal of Production Research, Taylor & Francis Journals, vol. 57(17), pages 5589-5603, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. M. R. Irshad & Muhammed Ahammed & R. Maya, 2025. "The Uniform Poisson–Ailamujia INAR(1) Process with Random Coefficient," Methodology and Computing in Applied Probability, Springer, vol. 27(2), pages 1-20, June.
    2. Shirozhan, M. & Bakouch, Hassan S. & Mohammadpour, M., 2023. "A flexible INAR(1) time series model with dependent zero-inflated count series and medical contagious cases," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 216-230.
    3. Wagner Barreto-Souza, 2019. "Mixed Poisson INAR(1) processes," Statistical Papers, Springer, vol. 60(6), pages 2119-2139, December.
    4. Jean Peyhardi, 2024. "Integer-valued autoregressive models based on quasi Pólya thinning operator," Statistical Inference for Stochastic Processes, Springer, vol. 27(3), pages 813-838, October.
    5. Marcelo Bourguignon & Christian H. Weiß, 2017. "An INAR(1) process for modeling count time series with equidispersion, underdispersion and overdispersion," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(4), pages 847-868, December.
    6. Ping Li & Feilong Lu, 2025. "Bayesian estimation of first-order integer generalized autoregressive models based on the negative binomial thinning operator," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 34(4), pages 865-894, September.
    7. Harvey, A., 2008. "Dynamic distributions and changing copulas," Cambridge Working Papers in Economics 0839, Faculty of Economics, University of Cambridge.
    8. Yang Lu, 2020. "A simple parameter‐driven binary time series model," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 39(2), pages 187-199, March.
    9. Fokianos, Konstantinos & Fried, Roland & Kharin, Yuriy & Voloshko, Valeriy, 2022. "Statistical analysis of multivariate discrete-valued time series," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    10. Jung, Robert C. & Kukuk, Martin & Liesenfeld, Roman, 2006. "Time series of count data: modeling, estimation and diagnostics," Computational Statistics & Data Analysis, Elsevier, vol. 51(4), pages 2350-2364, December.
    11. Brannas, Kurt, 1995. "Prediction and control for a time-series count data model," International Journal of Forecasting, Elsevier, vol. 11(2), pages 263-270, June.
    12. Snyder, Ralph D. & Ord, J. Keith & Beaumont, Adrian, 2012. "Forecasting the intermittent demand for slow-moving inventories: A modelling approach," International Journal of Forecasting, Elsevier, vol. 28(2), pages 485-496.
    13. Jian Pei & Yang Lu, 2025. "Forecasting natural disaster frequencies using nonstationary count time series models," Statistical Papers, Springer, vol. 66(3), pages 1-44, April.
    14. Youn Ahn, Jae & Jeong, Himchan & Lu, Yang, 2021. "On the ordering of credibility factors," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 626-638.
    15. Nobuhiko Terui & Masataka Ban, 2013. "Multivariate Time Series Model with Hierarchical Structure for Over-dispersed Discrete Outcomes," TMARG Discussion Papers 113, Graduate School of Economics and Management, Tohoku University, revised Aug 2013.
    16. Han, Yang & Liu, Zehao & Ma, Jun, 2020. "Growth cycles and business cycles of the Chinese economy through the lens of the unobserved components model," China Economic Review, Elsevier, vol. 63(C).
    17. Harvey, Andrew, 2010. "Tracking a changing copula," Journal of Empirical Finance, Elsevier, vol. 17(3), pages 485-500, June.
    18. Ana Julia Alves Camara & Valdério Anselmo Reisen & Glaura Conceicao Franco & Pascal Bondon, 2025. "Combining Generalized Linear Autoregressive Moving Average and Bootstrap Models for Analyzing Time Series of Respiratory Diseases and Air Pollutants," Mathematics, MDPI, vol. 13(5), pages 1-23, March.
    19. Svetunkov, Ivan & Boylan, John E., 2023. "iETS: State space model for intermittent demand forecasting," International Journal of Production Economics, Elsevier, vol. 265(C).
    20. Shang, Zuofeng, 2012. "On latent process models in multi-dimensional space," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1259-1266.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:compst:v:40:y:2025:i:2:d:10.1007_s00180-024-01511-3. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.