Research article

Power unit inverse Lindley distribution with different measures of uncertainty, estimation and applications

  • Received: 13 May 2024 Revised: 10 June 2024 Accepted: 21 June 2024 Published: 28 June 2024
  • MSC : 62F15, 62G20, 65C60

  • This paper introduced and investigated the power unit inverse Lindley distribution (PUILD), a novel two-parameter generalization of the famous unit inverse Lindley distribution. Among its notable functional properties, the corresponding probability density function can be unimodal, decreasing, increasing, or right-skewed. In addition, the hazard rate function can be increasing, U-shaped, or N-shaped. The PUILD thus takes advantage of these characteristics to gain flexibility in the analysis of unit data compared to the former unit inverse Lindley distribution, among others. From a theoretical point of view, many key measures were determined under closed-form expressions, including mode, quantiles, median, Bowley's skewness, Moor's kurtosis, coefficient of variation, index of dispersion, moments of various types, and Lorenz and Bonferroni curves. Some important measures of uncertainty were also calculated, mainly through the incomplete gamma function. In the statistical part, the estimation of the parameters involved was studied using fifteen different methods, including the maximum likelihood method. The invariant property of this approach was then used to efficiently estimate different uncertainty measures. Some simulation results were presented to support this claim. The significance of the PUILD underlying model compared to several current statistical models, including the unit inverse Lindley, exponentiated Topp-Leone, Kumaraswamy, and beta and transformed gamma models, was illustrated by two applications using real datasets.

    Citation: Ahmed M. Gemeay, Najwan Alsadat, Christophe Chesneau, Mohammed Elgarhy. Power unit inverse Lindley distribution with different measures of uncertainty, estimation and applications[J]. AIMS Mathematics, 2024, 9(8): 20976-21024. doi: 10.3934/math.20241021

    Related Papers:

  • This paper introduced and investigated the power unit inverse Lindley distribution (PUILD), a novel two-parameter generalization of the famous unit inverse Lindley distribution. Among its notable functional properties, the corresponding probability density function can be unimodal, decreasing, increasing, or right-skewed. In addition, the hazard rate function can be increasing, U-shaped, or N-shaped. The PUILD thus takes advantage of these characteristics to gain flexibility in the analysis of unit data compared to the former unit inverse Lindley distribution, among others. From a theoretical point of view, many key measures were determined under closed-form expressions, including mode, quantiles, median, Bowley's skewness, Moor's kurtosis, coefficient of variation, index of dispersion, moments of various types, and Lorenz and Bonferroni curves. Some important measures of uncertainty were also calculated, mainly through the incomplete gamma function. In the statistical part, the estimation of the parameters involved was studied using fifteen different methods, including the maximum likelihood method. The invariant property of this approach was then used to efficiently estimate different uncertainty measures. Some simulation results were presented to support this claim. The significance of the PUILD underlying model compared to several current statistical models, including the unit inverse Lindley, exponentiated Topp-Leone, Kumaraswamy, and beta and transformed gamma models, was illustrated by two applications using real datasets.


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    [1] D. V. Lindley, Fiducial distributions and Bayes' theorem, J. R. Stat. Soc., 20 (1958), 102–107. https://doi.org/10.1111/j.2517-6161.1958.tb00278.x doi: 10.1111/j.2517-6161.1958.tb00278.x
    [2] M. E. Ghitany, A. Barbra, S. Nadarajah, Lindley distribution and its application, Math. Comput. Simul., 78 (2008), 493–506. https://doi.org/10.1016/j.matcom.2007.06.007 doi: 10.1016/j.matcom.2007.06.007
    [3] M. Sankaran, The discrete Poisson-Lindley distribution, Biometrics, 26 (1970), 145–149.
    [4] M. Ghitany, D. Al-Mutairi, S. Nadarajah, Zero-truncated Poisson-Lindley distribution and its application, Math. Comput. Simul., 79 (2008), 279–287. https://doi.org/10.1016/j.matcom.2007.11.021 doi: 10.1016/j.matcom.2007.11.021
    [5] H. Zakerzadeh, A. Dolati, Generalized Lindley distribution, J. Math. Ext., 3 (2009), 13–25.
    [6] S. Nadarajah, H. S. Bakouch, R. Tahmasbi, A generalized Lindley distribution, Sankhya B, 73 (2011), 331–359. https://doi.org/10.1007/s13571-011-0025-9 doi: 10.1007/s13571-011-0025-9
    [7] M. Ghitany, F. Alqallaf, D. Al-Mutairi, H. A. Husain, A two-parameter weighted Lindley distribution and its applications to survival data, Math. Comput. Simul., 81 (2011), 1190–1201. https://doi.org/10.1016/j.matcom.2010.11.005 doi: 10.1016/j.matcom.2010.11.005
    [8] H. Bakouch, B. Al-Zahrani, A. Al-Shomrani, V. Marchi, F. Louzada, An extended Lindley distribution, J. Korean Stat. Soc., 41 (2012), 75–85. https://doi.org/10.1016/j.jkss.2011.06.002 doi: 10.1016/j.jkss.2011.06.002
    [9] W. Barreto-Souza, H. S. Bakouch, A new lifetime model with decreasing failure rate, Statistics, 47 (2013), 465–476. https://doi.org/10.1080/02331888.2011.595489 doi: 10.1080/02331888.2011.595489
    [10] R. Shanker, S. Sharma, R. Shanker, A two-parameter Lindley distribution for modeling waiting and survival times data, Appl. Math., 4 (2013), 363–368. https://doi.org/10.4236/am.2013.42056 doi: 10.4236/am.2013.42056
    [11] M. Ghitany, D. Al-Mutairi, N. Balakrishnan, L. Al-Enezi, Power Lindley distribution and associated inference, Comput. Stat. Data Anal., 64 (2013), 20–33. https://doi.org/10.1016/j.csda.2013.02.026 doi: 10.1016/j.csda.2013.02.026
    [12] A. Asgharzadeh, H. S. Bakouch, S. Nadarajah, F. Sharafi, A new weighted Lindley distribution with application, Braz. J. Probab. Stat., 30 (2016), 1–27. https://doi.org/10.1214/14-BJPS253 doi: 10.1214/14-BJPS253
    [13] M. Elgarhy, A. S. Hassan, S. Fayomi, Maximum likelihood and Bayesian estimation for two-parameter type I half logistic Lindley distribution, J. Comput. Theor. Nanos., 15 (2018), 3093–3101. https://doi.org/10.1166/jctn.2018.7600 doi: 10.1166/jctn.2018.7600
    [14] A. S. Hassan, R. E. Mohamed, M. Elgarhy, S. Alrajhi, On the alpha power transformed power Lindley distribution, J. Prob. Stat., 2019 (2019), 8024769. https://doi.org/10.1155/2019/8024769 doi: 10.1155/2019/8024769
    [15] V. K. Sharma, S. K. Singh, U. Singh, V. Agiwal, The inverse Lindley distribution: A stress-strength reliability model with application to head and neck cancer data, J. Ind. Prod. Eng., 32 (2015), 162–173. https://doi.org/10.1080/21681015.2015.1025901 doi: 10.1080/21681015.2015.1025901
    [16] A. M. Abd AL-Fattah, A. A. El-Helbawy, G. R. Al-Dayian, Inverted Kumaraswamy distribution: Properties and estimation, Pak. J. Stat., 33 (2017), 37–61.
    [17] K. V. P. Barco, J. Mazucheli, V. Janeiro, The inverse power Lindley distribution, Commun. Stat.-Simul. Comput., 46 (2017), 6308–6323. https://doi.org/10.1080/03610918.2016.1202274 doi: 10.1080/03610918.2016.1202274
    [18] A. S. Yadav, S. S. Maiti, M. Saha, The inverse xgamma distribution: Statistical properties and different methods of estimation, Ann. Data. Sci., 8 (2021), 275–293. https://doi.org/10.1007/s40745-019-00211-w doi: 10.1007/s40745-019-00211-w
    [19] S. Lee, Y. Noh, Y. Chung, Inverted exponentiated Weibull distribution with applications to lifetime data, Commun. Stat. Appl. Methods, 24 (2017), 227–240. https://doi.org/10.5351/CSAM.2017.24.3.227 doi: 10.5351/CSAM.2017.24.3.227
    [20] A. S. Hassan, M. Abd-Allah, On the inverse power Lomax distribution, Ann. Data. Sci., 6 (2019), 259–278. https://doi.org/10.1007/s40745-018-0183-y doi: 10.1007/s40745-018-0183-y
    [21] A. S. Hassan, R. E. Mohamed, Parameter estimation of inverse exponentiated Lomax with right censored data, Gazi Univ. J. Sci., 32 (2019), 1370–1386.
    [22] J. Y. Falgore, M. N. Isah, H. A. Abdulsalam, Inverse Lomax-Rayleigh distribution with application, Heliyon, 7 (2021), e08383. https://doi.org/10.1016/j.heliyon.2021.e08383 doi: 10.1016/j.heliyon.2021.e08383
    [23] M. H. Tahir, G. M. Cordeiro, S. Ali, S. Dey, A. Manzoor, The inverted Nadarajah-Haghighi distribution: Estimation methods and applications, J. Stat. Comput. Simul., 88 (2018), 2775–2798. https://doi.org/10.1080/00949655.2018.1487441 doi: 10.1080/00949655.2018.1487441
    [24] F. Louzada, P. L. Ramos, Nascimento, D. The inverse Nakagami-m distribution: A novel approach in reliability, IEEE Trans. Reliab., 67 (2018), 1030–1042. https://doi.org/10.1109/TR.2018.2829721 doi: 10.1109/TR.2018.2829721
    [25] A. S. Hassan, M. Elgarhy, R. Ragab, Statistical properties and estimation of inverted Topp-Leone distribution, J. Stat. Appl. Probab., 9 (2020), 319–331.
    [26] C. Chesneau, V. Agiwal, Statistical theory and practice of the inverse power Muth distribution, J. Comput. Math. Data Sci., 1 (2021), 100004. https://doi.org/10.1016/j.jcmds.2021.100004 doi: 10.1016/j.jcmds.2021.100004
    [27] M. H. Omar, S. Y. Arafat, M. P. Hossain, M. Riaz, Inverse Maxwell distribution and statistical process control: An efficient approach for monitoring positively skewed process, Symmetry, 13 (2021), 189. https://doi.org/10.3390/sym13020189 doi: 10.3390/sym13020189
    [28] N. Alsadat, M. Elgarhy, K. Karakaya, A. M. Gemeay, C. Chesneau, M. M. Abd El-Raouf, Inverse unit Teissier distribution: Theory and practical Examples, Axioms, 12 (2023), 502. https://doi.org/10.3390/axioms12050502 doi: 10.3390/axioms12050502
    [29] L. P. Sapkota, V. Kumar, Applications and some characteristics of inverse power Cauchy distribution, RT & A, 18 (2023), 301–315.
    [30] J. Mazucheli, A. F. B. Menezes, S. Dey, The unit Birnbaum-Saunders distribution with applications, Chil. J. Stat., 9 (2018), 47–57.
    [31] J. Mazucheli, A. F. B. Menezes, M. E. Ghitany, The unit Weibull distribution and associated inference, J. Appl. Probab. Stat., 13 (2018), 1–22.
    [32] J. Mazucheli, A. F. B. Menezes, L. B. Fernandes, R. P. de Oliveira, M. E. Ghitany, The unit Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates, J. Appl. Probab. Stat., 47 (2020), 954–974.
    [33] A. F. B. Menezes, J. Mazucheli, M. Bourguignon, A parametric quantile regression approach for modelling zero-or-one inflated double bounded data, The unit Weibull distribution and associated inference, Biometrical J., 63 (2021), 841–858.
    [34] J. Mazucheli, A. F. B. Menezes, S. Dey, Unit-Gompertz distribution with applications, Statistica, 79 (2019), 25–43.
    [35] J. Mazucheli, A. F. B. Menezes, S. Chakraborty, On the one parameter unit Lindley distribution and its associated regression model for proportion data, J. Appl. Stat., 46 (2019), 700–714. https://doi.org/10.1080/02664763.2018.1511774 doi: 10.1080/02664763.2018.1511774
    [36] M. E. Ghitany, J. Mazucheli, A. F. B. Menezes, F. Alqallaf, The unit-inverse Gaussian distribution: A new alternative to two parameter distributions on the unit interval, Commun. Stat. Theory Methods, 48 (2019), 3423–3438. https://doi.org/10.1080/03610926.2018.1476717 doi: 10.1080/03610926.2018.1476717
    [37] M. C. Korkmaz, C. Chesneau, On the unit Burr-XII distribution with the quantile regression modeling and applications, Comp. Appl. Math., 40 (2021), 29. https://doi.org/10.1007/s40314-021-01418-5 doi: 10.1007/s40314-021-01418-5
    [38] A.S. Hassan, A. Fayomi, A. Algarni, E. M. Almetwally, Bayesian and non-Bayesian inference for unit-exponentiated half-logistic distribution with data analysis, Appl. Sci., 12 (2022), 11253. https://doi.org/10.3390/app122111253 doi: 10.3390/app122111253
    [39] A. T. Ramadan, A. H. Tolba, B. S. El-Desouky, A unit half-logistic geometric distribution and its application in insurance, Axioms, 11 (2022), 676. https://doi.org/10.3390/axioms11120676 doi: 10.3390/axioms11120676
    [40] M. M. E. Abd El-Monsef, M. M. El-Awady, M. M. Seyam, A new quantile regression model for modelling child mortality, Int. J. Biomath., 10 (2022), 142–149.
    [41] A. Fayomi, A. S. Hassan, H. M. Baaqeel, E. M. Almetwally, Bayesian inference and data analysis of the unit-power Burr X distribution, Axioms, 12 (2023), 297. https://doi.org/10.3390/axioms12030297 doi: 10.3390/axioms12030297
    [42] A. S. Hassan, R. S. Alharbi, Different estimation methods for the unit inverse exponentiated Weibull distribution, Commun. Stat. Appl. Methods, 30 (2023), 191–213. https://doi.org/10.29220/CSAM.2023.30.2.191 doi: 10.29220/CSAM.2023.30.2.191
    [43] S. Nasiru, C. Chesneau, A. G. Abubakari, I. D. Angbing, Generalized unit half-logistic geometric distribution: Properties and regression with applications to insurance, Analytics, 2 (2023), 438–462. https://doi.org/10.3390/analytics2020025 doi: 10.3390/analytics2020025
    [44] C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., 27 (1948), 379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x doi: 10.1002/j.1538-7305.1948.tb01338.x
    [45] A. Rényi, On measures of entropy and information, Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, 1 (1960), 547–561.
    [46] L. L. Campbell, Exponential entropy as a measure of extent of a distribution, Z. Wahrscheinlichkeitstheorie verw Gebiete, 5 (1966), 217–225. https://doi.org/10.1007/BF00533058 doi: 10.1007/BF00533058
    [47] J. Havrda, F. Charvát, Quantification method of classification processes, concept of Structural $ a $-entropy, Kybernetika, 3 (1967), 30–35.
    [48] S. Arimoto, Information-theoretical considerations on estimation problems, Inf. Control, 19 (1971), 181–194. https://doi.org/10.1016/S0019-9958(71)90065-9 doi: 10.1016/S0019-9958(71)90065-9
    [49] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat Phys., 52 (1988), 479–487. https://doi.org/10.1007/BF01016429 doi: 10.1007/BF01016429
    [50] A. M. Awad, A. J. Alawneh, Application of entropy to a life-time model, IMA J. Math. Control Inf., 4 (1987), 143–148. https://doi.org/10.1093/imamci/4.2.143 doi: 10.1093/imamci/4.2.143
    [51] F. Lad, G. Sanfilippo, G. Agr, Extropy: Complementary dual of entropy, Statist. Sci., 30 (2015), 40–58. https://doi.org/10.1214/14-STS430 doi: 10.1214/14-STS430
    [52] N. Balakrishnan, F. Buono, M. Longobardi, On weighted extropies, Commun. Stat.-Theory Methods, 51 (2022), 6250–6267. https://doi.org/10.1080/03610926.2020.1860222
    [53] D. P. Murthy, M. Xie, R. Jiang, Weibull models, New York: John Wiley & Sons, 2004.
    [54] A. Krishna, R. Maya, C. Chesneau, M. R. Irshad, The unit Teissier distribution and its applications, Math. Comput. Appl., 27 (2022), 12. https://doi.org/10.3390/mca27010012 doi: 10.3390/mca27010012
    [55] A. Pourdarvish, S. M. T. K. Mirmostafaee, K. Naderi, The exponentiated Topp-Leone distribution: Properties and application, J. Appl. Environ. Biol. Sci., 5 (2015), 251–256.
    [56] A. Grassia, On a family of distributions with argument between 0 and 1 obtained by transformation of the gamma and derived compound distributions, Austral. J. Statist., 19 (1977), 108–114. https://doi.org/10.1111/j.1467-842X.1977.tb01277.x doi: 10.1111/j.1467-842X.1977.tb01277.x
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