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The balanced discrete triplet Lindley model and its INAR(1) extension: properties and COVID-19 applications

  • Masoumeh Shirozhan ORCID logo EMAIL logo , Naushad A. Mamode Khan and Célestin C. Kokonendji
Published/Copyright: November 24, 2022
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The International Journal of Biostatistics
From the journal The International Journal of Biostatistics Volume 19 Issue 2

Abstract

This paper proposes a new flexible discrete triplet Lindley model that is constructed from the balanced discretization principle of the extended Lindley distribution. This model has several appealing statistical properties in terms of providing exact and closed form moment expressions and handling all forms of dispersion. Due to these, this paper explores further the usage of the discrete triplet Lindley as an innovation distribution in the simple integer-valued autoregressive process (INAR(1)). This subsequently allows for the modeling of count time series observations. In this context, a novel INAR(1) process is developed under mixed Binomial and the Pegram thinning operators. The model parameters of the INAR(1) process are estimated using the conditional maximum likelihood and Yule-Walker approaches. Some Monte Carlo simulation experiments are executed to assess the consistency of the estimators under the two estimation approaches. Interestingly, the proposed INAR(1) process is applied to analyze the COVID-19 cases and death series of different countries where it yields reliable parameter estimates and suitable forecasts via the modified Sieve bootstrap technique. On the other side, the new INAR(1) with discrete triplet Lindley innovations competes comfortably with other established INAR(1)s in the literature.

Keywords: conditional maximum likelihood; COVID-19; INAR(1); sieve bootstrap; triplet Lindley
Corresponding author: Masoumeh Shirozhan, Water and Wastewater Company, Ardabil, Ardabil Province, Iran, E-mail:

Acknowledgements

The authors sincerely thank an Associate Editor and two anonymous referees for their valuable comments that improved the paper. For the third coauthor, this work is supported by the EIPHI Graduate School (contract ANR-17-EURE-0002).

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

A.1 The proof of Proposition 2.2

  1. The conditional expectation of the process is obtained as follows

    E X t | X t − 1 = α ϕ X t − 1 + E Z , E X t | X t − 2 = E E X t | X t − 1 | X t − 2 = α ϕ 2 X t − 2 + 1 + α ϕ E Z .

    By induction, we can conclude that

    E ( X t | X t − k ) = ( α ϕ ) k X t − k + ( 1 − ( α ϕ ) k ) E ( Z ) 1 − ( α ϕ ) ,

    which is a linear function in X t .

  2. The second order conditional expectation of the BL-MINAR(1) process is computed as

    E X t 2 | X t − 1 = α 2 ϕ X t − 1 2 + α ϕ 1 − α X t − 1 + 2 α ϕ E Z X t − 1 + E Z 2 ,

    and

    E X t 2 | X t − 2 = α 4 ϕ 2 X t − 2 2 + α 2 ϕ 2 1 − α 2 X t − 2 + 2 α 2 ϕ 2 1 + α E Z X t − 2 + 1 + α 2 ϕ E Z 2 + α ϕ 1 − α E Z + 2 α ϕ E 2 Z ,

    subsequently,

    E X t 2 | X t − 3 = α 6 ϕ 3 X t − 3 2 + α 3 ϕ 3 1 − α 1 + α + α 2 X t − 3 + 2 α 3 ϕ 3 1 + α + α 2 E Z X t − 3 + 1 + α 2 ϕ + α 4 ϕ 2 E Z 2 + α ϕ 1 − α 1 + α ϕ + α 2 ϕ E Z + 2 α ϕ 1 + α ϕ + α 2 ϕ E 2 Z .

    By induction, we can conclude that

    E X t 2 | X t − k = α 2 k ϕ k X t − k 2 + 2 α ϕ k E Z X t − k ∑ i = 0 k − 1 α i + α ϕ k 1 − α X t − k ∑ i = 0 k − 1 α i + E Z 2 ∑ i = 0 k − 1 α 2 ϕ i + α ϕ ( 1 − α ) E ( Z ) ∑ i = 0 k − 1 α ϕ i ∑ j = 0 i α j + 2 α ϕ E 2 ( Z ) ∑ i = 0 k − 1 α ϕ i ∑ j = 0 i α j .

    By the variance definition and some elementary calculations, the proof is completed.

  3. The autocovariance is computed as follows

    C o v ( X t , X t − 1 ) = ϕ C o v ( α ◦ X t − 1 + Z t , X t − 1 ) + ( 1 − ϕ ) C o v ( Z t , X t − 1 ) = ϕ C o v ( α ◦ X t − 1 , X t − 1 ) = ϕ E ( α ◦ X t − 1 X t − 1 ) − E ( α ◦ X t − 1 ) E ( X t − 1 ) = ϕ α E X t − 1 2 − E 2 ( X t − 1 ) = α ϕ Var ( X ) .

Consequently, for lag 2,

ρ ( 2 ) = C o v ( X t , X t − 2 ) = ϕ C o v ( α ◦ X t − 1 + Z t , X t − 2 ) + ( 1 − ϕ ) C o v ( Z t , X t − 2 ) = ϕ C o v ( α ◦ X t − 1 , X t − 2 ) = α ϕ C o v ( α ◦ X t − 1 , X t − 2 ) = α ϕ 2 Var ( X ) .

Hence, by induction, the autocovariance of lag k is concluded as Cov(X t , X t−k ) = (αϕ) k  Var(X). Therefore, the autocorrelation function is obtained as ρ(k) = Corr(X t , X t−k ) = (αϕ) k

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Received: 2022-01-05
Accepted: 2022-10-26
Published Online: 2022-11-24

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijb-2022-0001
Keywords for this article
conditional maximum likelihood; COVID-19; INAR(1); sieve bootstrap; triplet Lindley

Articles in the same Issue

  1. Frontmatter
  2. Part-1: SMAC 2021 Webconference
  3. Statistics, philosophy, and health: the SMAC 2021 webconference
  4. Part-2: Regular Articles
  5. “Show me the DAG!”
  6. Causal inference for oncology: past developments and current challenges
  7. The EBM+ movement
  8. Bayesianism from a philosophical perspective and its application to medicine
  9. Bayesian inference for optimal dynamic treatment regimes in practice
  10. Agent-based modeling in medical research, virtual baseline generator and change in patients’ profile issue
  11. Agent based modeling in health care economics: examples in the field of thyroid cancer
  12. A copula-based set-variant association test for bivariate continuous, binary or mixed phenotypes
  13. Detection of atypical response trajectories in biomedical longitudinal databases
  14. Potential application of elastic nets for shared polygenicity detection with adapted threshold selection
  15. Error analysis of the PacBio sequencing CCS reads
  16. A SIMEX approach for meta-analysis of diagnostic accuracy studies with attention to ROC curves
  17. Statistical modelling of COVID-19 and drug data via an INAR(1) process with a recent thinning operator and cosine Poisson innovations
  18. The balanced discrete triplet Lindley model and its INAR(1) extension: properties and COVID-19 applications
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