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Generalized inverted Kumaraswamy generated family of distributions: theory and applications. (English) Zbl 1516.62355

J. Appl. Stat. 46, No. 16, 2927-2944 (2019); correction ibid. 49, No. 6, 1611 (2022).
Summary: This paper proposes a new generator function based on the inverted Kumaraswamy distribution and introduces ‘generalized inverted Kumaraswamy-G’ family of distributions. We provide a comprehensive account of some of its mathematical properties that include the ordinary and incomplete moments, quantile and generating functions and order statistics. The infinite mixture representations for probability density and cumulative distribution and entropy functions of the new family are also established. The density function of the \(i\)th-order statistics is expressed as an infinite linear combination of baseline densities and model parameters are estimated by maximum likelihood method. Four special models of this family are also derived along with their respective hazard rate functions. The maximum likelihood estimation (MLE) method is used to obtain the model parameters. Monte Carlo simulation experiments are executed to assess the performance of the ML estimators under the corresponding generated models while some data applications are also illustrated. The results of the study show that the proposed distribution is more flexible as compared to the baseline model. This distribution especially can be used to model symmetric, left-skewed, right-skewed and reversed-J data sets.

MSC:

62-XX Statistics
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[1] Abd Al-Fattah, A. M.A.; EL-Helbawy, A. A.; AL-Dayian, G. R., Inverted Kumaraswamy distribution: properties and estimation, Pak. J. Stat. Oper. Res., 33, 37-67 (2017)
[2] Abd El-Kader, R. I.; AL-Dayian, G. R.; AL-Gendy, S. A., Inverted pareto type I distribution: properties and estimation, J. Fac. Commer. AL-Azhar Univ., 21, 19-40 (2003)
[3] Al-Dayian, G. R., Burr type III distribution: properties and estimation, Egypt. Stat. J., 43, 102-116 (1999)
[4] Alexander, C.; Cordeiro, G. M.; Ortega, E. M.M.; Sarabia, J. M., Generalized beta generated distributions, Comput. Stat. Data Anal., 56, 1880-1897 (2012) · Zbl 1245.60015 · doi:10.1016/j.csda.2011.11.015
[5] Alzaatreh, A.; Lee, C.; Famoye, F., A new method for generating families of continuous distributions, Metron, 71, 63-79 (2013) · Zbl 1302.62026 · doi:10.1007/s40300-013-0007-y
[6] Bourguignon, M.; Silva, R. B.; Cordeiro, G. M., The Weibull-G family of probability distributions, J. Data Sci., 12, 53-68 (2014)
[7] Calabria, R.; Pulcini, G., On the maximum likelihood and least squares estimation in the inverse Weibull distribution, J. Stat. Appl., 2, 53-66 (1990)
[8] Cordeiro, G. M.; de Castro, M., A new family of generalized distributions, J. Stat. Comput. Simul., 81, 883-893 (2011) · Zbl 1219.62022 · doi:10.1080/00949650903530745
[9] Cordeiro, G. M.; Ortega, E. M.M.; da Cunha, D. C.C., The exponentiated generalized class of distribution, J. Data Sci., 11, 1-27 (2013)
[10] Cordeiro, G. M.; Ortega, E. M.M.; Bozidar, P. V.; Pescim, R. R., The Lomax generator of distributions: properties, minification process and regression model, Appl. Math. Comput., 247, 465-486 (2014) · Zbl 1338.60031
[11] Elgarhy, M.; Hassan, A. S.; Rashed, M., Garhy-generated family of distributions with application, Math. Theory Model., 6, 1-15 (2016)
[12] Elgarhy, M.; Haq, M. A.; Ozel, G.; Nasir, A., A new exponentiated extended family of distributions with applications, Gazi Univ. J. Sci., 30, 101-115 (2017)
[13] Eugene, N.; Lee, C.; Famoye, F., Beta-normal distribution and its applications, Commun. Stat. Theory Methods, 31, 497-512 (2002) · Zbl 1009.62516 · doi:10.1081/STA-120003130
[14] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (2014), Elsevier Academic Press, Oxford · Zbl 0918.65002
[15] Hassan, A. S.; Elgarhy, M., Kumaraswamy Weibull-generated family of distributions with applications, Adv. Appl. Stat., 48, 205-239 (2016) · Zbl 1391.62021
[16] Hassan, A. S.; Elgarhy, M., A new family of exponentiated Weibull-generated distributions, Int. J. Math. Appl., 4, 135-148 (2016)
[17] Hassan, A. S.; Hemeda, S. E., The additive Weibull-G family of probability distributions, Int. J. Math. Appl., 4, 151-164 (2016)
[18] Hassan, A. S.; Elgarhy, M.; Shakil, M., Type II half logistic family of distributions with applications, Pak. J. Stat. Oper. Res., 13, 245-264 (2017) · Zbl 1509.60041 · doi:10.18187/pjsor.v13i2.1560
[19] Hassan, A. S.; Nassr, S. G., Power Lindley-G family, Ann. Data Sci., 6, 1-22 (2018)
[20] Hassan, A. S.; Nassr, S. G., The inverse Weibull generator of distributions: properties and applications, J. Data Sci., 16, 732-742 (2018)
[21] Jones, M. C., Families of distributions arising from the distributions of order statistics, Test, 13, 1-43 (2004) · Zbl 1110.62012 · doi:10.1007/BF02602999
[22] Nichols, M. D.; Padgett, W. J., A bootstrap control chart for Weibull percentiles, Qual. Reliab. Eng. Int., 22, 141-151 (2006) · doi:10.1002/qre.691
[23] Prakash, G., Inverted exponential distribution under a Bayesian view point, J. Mod. Appl. Stat. Methods., 11, 190-202 (2012) · doi:10.22237/jmasm/1335845700
[24] Reǹyi, A., On measures of entropy and information. Proceedings of the 4th Fourth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1961, pp. 547-561. · Zbl 0106.33001
[25] Ristic‘, M. M.; Balakrishnan, N., The gamma-exponentiated exponential distribution, J. Stat. Comput. Simul., 82, 1191-1206 (2012) · Zbl 1297.62033 · doi:10.1080/00949655.2011.574633
[26] Shannon, C. E., A mathematical theory of communication, Bell Syst. Tech. J., 27, 379-432 (1948) · Zbl 1154.94303 · doi:10.1002/j.1538-7305.1948.tb01338.x
[27] Tahir, M. H.; Cordeiro, G. M.; Alzaatreh, A.; Mansoor, M.; Zubair, M., The logistic-X family of distributions and its applications, Commun. Stat. Theory Methods, 45, 7326-7349 (2016) · Zbl 1349.60017 · doi:10.1080/03610926.2014.980516
[28] Torabi, H.; Montazari, N. H., The gamma-uniform distribution and its application, Kybernetika, 48, 16-30 (2012) · Zbl 1243.93123
[29] Torabi, H.; Montazari, N. H., The logistic-uniform distribution and its application, Commun. Stat. Simul. Comput., 43, 2551-2569 (2014) · Zbl 1462.62110 · doi:10.1080/03610918.2012.737491
[30] Zografos, K.; Balakrishnan, N., On families of beta- and generalized gamma-generated distributions and associated inference, Stat. Methodol., 6, 344-362 (2009) · Zbl 1463.62023 · doi:10.1016/j.stamet.2008.12.003
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