Confidence Intervals for the Zeghdoudi Distribution Parameter: Applications in Precipitation and COVID-19 Data Analysis

Benjamas  Tulyanitikul; Wararit Panichkitkosolkul

doi:10.59796/jcst.V15N1.2025.87

Authors

Benjamas Tulyanitikul Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 12120, Thailand
Wararit Panichkitkosolkul Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 12120, Thailand

DOI:

https://doi.org/10.59796/jcst.V15N1.2025.87

Keywords:

lifetime distribution, interval estimation, likelihood, Wald, bootstrap

Abstract

This paper proposes four confidence intervals (CIs) for estimating the parameters of the Zeghdoudi distribution, which is commonly used in the analysis of lifetime data. We introduce and evaluate the likelihood-based, Wald-type, bootstrap-t, and bias-corrected and accelerated (BCa) bootstrap CIs using Monte Carlo simulation studies and apply them to two real datasets. We assess the effectiveness of these CIs by evaluating their empirical coverage probability (CP) and average length (AL), which offer valuable insights into their performance in various scenarios. Furthermore, we have developed an explicit formulation of the Wald-type’s CI formula, simplifying its computation. The results demonstrate that the CPs of likelihood-based and Wald-type CIs converge towards the nominal confidence level of 0.95 for all cases. However, when the sample size is small, the bootstrap-t and BCa bootstrap CIs have CPs less than 0.95. On the other hand, as sample sizes increase, the CPs of all CIs tend to approach 0.95. However, when the sample sizes are small, the CPs of the bootstrap-t and BCa bootstrap CIs tend to decrease. We verified the efficacy of CIs by applying them to precipitation data and COVID-19 mortality rate data, and the results matched those from the simulation study.

References

Almongy, H. M., Almetwally, E. M., Aljohani, H. M., Alghamdi, A. S., & Hafez, E. H. (2021). A new extended Rayleigh distribution with applications to COVID-19 data. Results in Physics, 23, Article 104012. https://doi.org/10.1016/j.rinp.2021.104012

Al-Omari, A., & Shraa, D. (2019). Darna distribution: Properties and application. Electronic Journal of Applied Statistical Analysis, 12(2), 520-541. https://doi.org/10.1285/i20705948v12n2p520

Alzoubi, L., Gharaibeh, M., Alkhazaalh, A., & Benrabia, M. (2022a). Loai distribution: Properties, parameters estimation, and application to COVID-19 real data. Mathematics and Statistics in Engineering Applications, 71(4), 1231-1255. https://doi.org/10.17762/msea.v71i4.619

Alzoubi, L., Gharaibeh, M., Al-Khazaleh, A., & Benrabia, M. (2022b). Estimation of Sameera distribution parameters with applications to real data. Applied Mathematics & Information Sciences, 16(6), 1057-1071. http://dx.doi.org/10.18576/amis/160624

Bolker, B. M. (2023). bbmle: Tools for general maximum likelihood estimation (Version 1.0.25.1) [Computer software]. Retrieved from https://CRAN.R-project.org/package=bbmle

Canty, A., & Ripley, B. (2024). boot: Bootstrap functions (Version 1.3-30) [Computer software]. Retrieved from https://cran.r-project.org/package=boot

Dey, S., Ali, S., & Park, C. (2015). Weighted exponential distribution: properties and different methods of estimation. Journal of Statistical Computation and Simulation, 85(18), 3641-3661. https://doi.org/10.1080/00949655.2014.992346

Elechi, O., Okereke, E. W., Chukwudi, I. H., Chizoba, K. L., & Wale, O. T. (2022). Iwueze’s distribution and its application. Journal of Applied Mathematics and Physics, 10(12), 3783-3803. https://doi.org/10.4236/jamp.2022.1012251

Gharaibeh, M. (2020). Transmuted Aradhana distribution: Properties and application. Jordan Journal of Mathematics and Statistics (JJMS), 13(2), 287-304.

Gharaibeh, M. M. (2021). Gharaibeh distribution and its applications. Journal of Statistics Applications & Probability, 10(2), 441-452. https://doi.org/10.18576/jsap/100214

Ghitany, M., Al-Mutairi, D., Balakrishnan, N., & Alenezi, L. (2013). Power Lindley distribution and associated inference. Computational Statistics & Data Analysis, 64, 20-33. https://doi.org/10.1016/j.csda.2013.02.026

Henningsen, A., & Toomet, O. (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics, 26(3), 443-458. https://doi.org/10.1007/s00180-010-0217-1

Hinkley, D. (1977). On quick choice of power transformation. Journal of the Royal Statistical Society: Series C (Applied Statistics), 26(1), 67-69. https://doi.org/10.2307/2346869

Kiusalaas, J. (2013). Numerical Methods in Engineering with Python 3. Cambridge, UK: Cambridge University Press.

Kostyshak, S. (2024). bootstrap: Functions for the Book “An Introduction to the Bootstrap (Version 2019.6) [Computer software]. Retrieved from https://cran.r-project.org/web/packages/bootstrap

Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, 20(1), 102-107. https://doi.org/10.1111/j.2517-6161.1958.tb00278.x

Manoj, K., & Elangovan, R. (2020). Weighted Odoma distribution with properties and applications in cancer data. Science & Technology Development, 8, 159-174.

Mbegbu, J. I., & Echebiri, U. V. (2022). Juchez probability distribution: Properties and applications. Asian Journal of Probability and Statistics, 20(2), 56-71. https://doi.org/10.9734/AJPAS/2022/v20i2419

Merovci, F. (2013a). Transmuted Lindley distribution. Journal of Open Problems in Computer Science and Mathematics, 6(2), 63-72. https://doi.org/10.12816/0006170

Merovci, F. (2013b). Transmuted Rayleigh distribution. Austrian Journal of Statistics, 42(1), 21-31. https://doi.org/10.17713/ajs.v42i1.163

Messaadia, H., & Zeghdoudi, H. (2018). Zeghdoudi distribution and its applications. International Journal of Computational Science and Mathematics, 9(1), 58–65. https://doi.org/10.1504/IJCSM.2018.090722

Mohiuddin, M., Dar, S. A., Khan, A. A., & Ahajeeth, M. (2022). On weighted Nwikpe distribution: Properties and applications. Information Sciences Letters, 11(1), 85-96. https://doi.org/10.18576/isl/110110

Nwry, A. W., Kareem, H. M., Ibrahim, R. B., & Mohammed, S. M. (2021). Comparison between bisection, Newton, and secant methods for determining the root of the non-linear equation using MATLAB. Turkish Journal of Computer and Mathematics Education, 12(14), 1115-1122. https://doi.org/10.17762/turcomat.v12i14.10397

Ola, A.-T., & Mohammed, M. G. (2023). Ola distribution: A new one parameter model with applications to engineering and COVID-19 data. Applied Mathematics & Information Sciences, 17(2), 242-252.

Panichkitkosolkul, W. (2024). Non-parametric bootstrap confidence intervals for index of dispersion of zero-truncated Poisson-Lindley distribution. Maejo International Journal of Science and Technology, 18(1), 1-12.

Pawitan, Y. (2001). All Likelihood: Statistical Modelling and Inference Using Likelihood. Clarendon Press.

Severini, T. A. (2000). Likelihood Methods in Statistics. Oxford University Press.

Shanker, R. (2015a). Shanker distribution and its applications. International Journal of Statistics and Applications, 5(6), 338-348. https://doi.org/10.5923/j.statistics.20150506.08

Shanker, R. (2015b). Akash distribution and its applications. International Journal of Probability and Statistics, 4(3), 65-75. https://doi.org/10.5923/j.ijps.20150403.01

Shanker, R. (2016a). Aradhana distribution and its applications. International Journal of Statistics and Applications, 6(1), 23-34. https://doi.org/10.5923/j.statistics.20160601.04

Shanker, R. (2016b). Sujatha distribution and its applications. Statistics in Transition New Series, 17(3), 391–410. https://doi.org/10.21307/stattrans-2016-029

Shanker, R. (2016c). Amarendra distribution and its applications. American Journal of Mathematics and Statistics, 6(1), 44-56. https://doi.org/10.5923/j.ajms.20160601.05

Shanker, R. (2016d). Devya distribution and its applications. International Journal of Statistics and Applications, 6(4), 189-202. https://doi.org/10.5923/j.statistics.20160604.01

Shanker, R. (2016e). Garima distribution and its application to model behavioral science data. Biometrics & Biostatistics International Journal, 4(1), 275-281. https://doi.org/10.15406/bbij.2016.04.00116

Shanker, R. (2017a). Rama distribution and its application. International Journal of Statistics and Applications, 7(1), 26-35. https://doi.org/10.5923/j.statistics.20170701.04

Shanker, R. (2017b). Akshaya distribution and its application. American Journal of Applied Mathematics and Statistics, 7(2), 51-59. https://doi.org/10.5923/j.ajms.20170702.01

Shanker, R. (2017c). Rani distribution and its application. Biometrics & Biostatistics International Journal, 6(1), 256-265. https://doi.org/10.15406/bbij.2017.06.00155

Shanker, R. (2017e). Suja distribution and its application. International Journal of Probability and Statistics, 6(2), 11-19. https://doi.org/10.5923/j.ijps.20170602.01

Shanker, R., & Shukla, K. K. (2017a). A generalization of Sujatha distribution and its applications with real lifetime data. Journal of Institute of Science and Technology, 22(1), 66-83. https://doi.org/10.3126/jist.v22i1.17742

Shanker, R., & Shukla, K. K. (2017b). Ishita distribution and its applications. Biometrics & Biostatistics International Journal, 5(2), 1-9. https://doi.org/10.15406/bbij.2017.05.00126

Shanker, R., & Shukla, K. K. (2018). Power Ishita distribution and its applications to model lifetime data. Statistics in Transition New Series, 19(11), 453-466. https://doi.org/10.21307/stattrans-2018-008

Shanker, R., & Shukla, K. K. (2023). Komal distribution with properties and application in survival analysis. Biometrics & Biostatistics International Journal, 12(2), 40-44. https://doi.org/10.15406/bbij.2023.12.00381

Shanker, R., Shukla, K. K., Ranjan, A., & Shanker, R. (2021). Adya distribution with properties and application. Biometrics & Biostatistics International Journal, 10(3), 81-88. https://doi.org/10.15406/bbij.2021.10.00334

Shukla, K. K. (2018a). Prakaamy distribution with properties and applications. Journal of Applied Quantitative Methods, 13(3), 30-38.

Shukla, K. K. (2018b). Pranav distribution with properties and its applications. Biometrics & Biostatistics International Journal, 7(3), 244-254. https://doi.org/10.15406/bbij.2018.07.00215

Shukla, K. K., & Shanker, R. (2020). A weighted Pranav distribution and its application to survival data. Indian Journal of Industrial and Applied Mathematics, 11(1), 79-90. https://doi.org/10.5958/1945-919X.2020.00006.7

Wasinrat, S., & Choopradit, B. (2023). The Poisson inverse Pareto distribution and its application. Thailand Statistician, 21(1), 110-124.