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Al-Bahir Journal for Engineering and Pure Sciences

Abstract

The literature contains innumerable probability distributions for modeling over-dispersion and under-dispersion count datasets from various fields of study. However, some of these proposed distributions are inadequate due to empirical or theoretical characteristics. Therefore, minimizing information loss during modeling has prompted the demand to modify the classical discrete distributions. A new two-parameter count distribution is proposed by combining Poisson and Lindley-quasi XGamma distributions via a continuous mixture technique. Some statistical properties have been derived and studied, including factorial moments, raw moments, probability generating function, moment generating function, characteristic function, mean, variance, dispersion index, skewness, and kurtosis. The shape of the PMF and dispersion index suggest that the proposed distribution is right-skewed with a heavy tail, over-dispersion, and approximately equi-dispersion. The unknown parameters of the proposed model are estimated using both maximum likelihood and Bayesian techniques. The usefulness and flexibility of the proposed distribution are measured using two distinctive datasets. The application results reveal that the developed distribution provides maximum fit to the given datasets compared to the other eight standard discrete distributions. The Poisson Lindley-quasi XGamma distribution should therefore be considered by researchers when modeling over-dispersed count data from all fields of study.

References

[1] Wongrin W, Bodhisuwan W. The Poisson-generalised Lindley distribution and its applications. Songklanakarin J Sci Technol 2016;38(6).

[2] Para BA, Jan TR, Bakouch HS. Poisson Xgamma distribution: a discrete model for count data analysis. Model Assisted Statistics Appl 2020;15(2):139e51.

[3] Ahmad PB, Wani MK. A new compound distribution and its applications in over-dispersed count data. Annals of Data Science 2023:1e22.

[4] Shanker R, Shukla KK, Leonida TA. A two-parameter Poisson-Akash distribution with properties and applications. Int J Probab Stat 2018;7(4):114e23.

[5] Bereta EM, Louzanda F, Franco MA. The Poisson-Weibull distribution. Adv Appl Stat 2011;22(2):107e18.

[6] Chesneau C, Bakouch H, Akdogan Y, Karakaya K. The binomial-discrete Poisson-Lindley model: modeling and applications to count regression. Commun Math Res 2022; 38(1):28e51

. [7] Altun E, Bhati D, Khan NM. A new approach to model the counts of earthquakes: INARPQX (1) process. SN Appl Sci 2021;3:1e17.

[8] Maya R, Irshad MR, Chesneau C, Nitin SL, Shibu DS. On discrete PoissoneMirra distribution: regression, INAR (1) process and applications. Axioms 2022;11(5):193.

[9] Alkhairy I. Classical and Bayesian inference for the discrete Poisson Ramos-Louzada distribution with application to COVID-19 data. Math Biosci Eng 2023;20(8):14061e80.

[10] Grine R, Zeghdoudi H. On Poisson quasi-Lindley distribution and its applications. J Mod Appl Stat Methods 2017; 16(2):21.

[11] Shukla KK, Shanker R, Tiwari MK, Ababneh F. Size-biased Poisson-pranav distribution and its applications. Intern J Agri Stat Sci 2023;19(2).

[12] Irshad MR, Aswathy S, Maya R, Nadarajah S. New oneparameter over-dispersed discrete distribution and its Fig. 11. Autocorrelation plots of the posterior parameters for dataset II. 94 AL-BAHIR JOURNAL FOR ENGINEERING AND PURE SCIENCES 2025;6:80e95 application to the nonnegative integer-valued autoregressive model of order one. Mathematics 2023;12(1):81.

[13] Chesneau C, D’cruz V, Maya R, Irshad M. A novel discrete distribution based on the mixture of Poisson and sum of two Lindley random variables. REVSTAT-Stat Journal 2023.

[14] Karakaya K. A new discrete distribution with applications to radiation, smoking, and health data. J Rad Res Appl Sci 2023; 16(4):100735.

[15] Alghamdi SM, Albalawi O, Almarzouki SM, Nagarjuna VB, Nasiru S, Elgarhy M. Different estimation methods of the modified Kies Topp-Leone model with applications and quantile regression. PLoS One 2024;19(9):e0307391.

[16] Hassan A, Wani SA, Shafi S. Lindley-quasI xgamma distribution: properties and applications. Pak J Statist 2020;36(1): 73e89.

[17] Team RC. R: a language and environment for statistical computing. R Foundation for Statistical Computing; 2020.

[18] Shukla KK, Shanker R, Tiwari MK. A new one parameter discrete distribution and its applications. J Stat Manag Syst 2022;25(1):269e83.

[19] Shukla KK, Shanker R. The Poisson-Prakaamy distribution and its applications. Aligarh J Stat 2020;40:137e50.

[20] Eldeeb AS, Ahsan-Ul-Haq M, Babar A. A discrete analog of inverted Topp-Leone distribution: properties, estimation and applications. Intern J Analysis Applications 2021;19(5): 695e708.

[21] Hussain T, Ahmad M. Discrete inverse Rayleigh distribution. P J Stat 2014;30(2).

[22] Borbye S, Nasiru S, Ajongba KK. Poisson XRani distribution: an alternative discrete distribution for overdispersed count data. Int J Math Math Sci 2024;2024(1):5554949.

[23] Gillariose J, Balogun OS, Almetwally EM, Sherwani RAK, Jamal F, Joseph J. On the discrete Weibull MarshalleOlkin family of distributions: properties, characterizations, and applications. Axioms 2021;10(4):287.

[24] Almetwally EM, Ibrahim GM. Discrete alpha power inverse Lomax distribution with application of COVID-19 data. Int J Appl Math 2020;9(6):11e22.

[25] Afify AZ, Elmorshedy M, Eliwa M. A new skewed discrete model: properties, inference, and applications. Pak J Statistics Oper Res 2021:799e816.

[26] Su YS, Yajima M. R2jags: a package for running jags from R. R package version 0.03-08. 2012

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