The present study has shown the existence of a novel category of distributions known as the Burr XII general (BXII-G) distribution. The ordinary and incomplete moments, quantile function, entropies (Rényi and Shannon), and order statistics are derived, yielding explicit formulas. The maximum likelihood estimators for the parameters are thoroughly examined. The present discourse concerns examining the Burr XII-log-logistic (BXII-LL) distribution which constitutes a distinctive model within the newly introduced family. Three empirical data sets from real-life scenarios are analyzed to provide a practical demonstration of the suggested family.

Burr XII distribution, maximum likelihood estimation, ordinary and incomplete moments, entropies, quantile function.

Received: October 12, 2023;  Accepted: December 29, 2023; Published: January 13, 2024

How to cite this article: Ali A. Al-Shomrani and Ahmed I. Shawky, A new family of distributions: properties and applications, Advances and Applications in Statistics 91(3) (2024), 257-300. http://dx.doi.org/10.17654/0972361724015

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] M. Aboray and N. S. Butt, Extended Weibull Burr XII distribution: properties and applications, Pak. J. Stat. Oper. Res. 15(4) (2019), 891-903.
[2] H. Akaike, Information theory and an extension of the maximum likelihood principle, Second International Symposium on Information Theory, B. N. Petrov and F. Csaki, eds., Akadémiai Kiadó, Budapest, 1973, pp. 267-281.
[3] H. Akaike, A new look at the statistical model identification, IEEE Trans. Automat. Control 19 (1974), 716-723.
[4] H. Akaike, Likelihood of a model and information criteria, J. Econometrics 16 (1981), 3-14.
[5] O. M. Akpa and E. I. Unuabonah, Small-sample corrected Akaike information criterion: an appropriate statistical tool for ranking of adsorption isotherm models, Desalination 272 (2011), 20-26.
[6] C. Alexander, G. M. Cordeiro, E. M. M. Ortega and J. M. Sarabia, Generalized beta generated distributions, Comput. Statist. Data Anal. 56 (2012), 1880-1897.
[7] A. Y. Al-Saiari, L. A. Baharith and S. A. Mousa, Marshall-Olkin extended Burr Type XII distribution, International Journal of Statistics and Probability 3(1) (2014), 78-84.
[8] A. Al-Shomrani, O. Arif, A. I. Shawky, S. Hanif and M. Q. Shahbaz, Topp-Leone family of distribution: some properties and application, Pak. J. Stat. Oper. Res. 12 (2016), 443-451.
[9] A. Alzaatreh, C. Lee and F. Famoye, A new method for generating families of continuous distributions, Metron 71 (2013), 63-79.
[10] A. Alzaghal, F. Famoye and C. Lee, Exponentiated T-X family of distributions with some applications, International Journal of Statistics and Probability 2 (2013), 31-49.
[11] M. Amini, S. M. T. K. MirMostafee and J. Ahmadi, Log-gamma-generated families of distributions, Statistics 48 (2012), 913-932.
[12] D. R. Anderson, K. P. Burhnam and G. C. White, Comparison of Akaike information criterion and consistent Akaike information criterion for model selection and statistical inference from capture-recapture studies, J. Appl. Stat. 25(2) (1998), 263-282.
[13] E. G. Antonio, C. Q. da-Silva and G. M. Cordeiro, Two extended Burr models: theory and practice, Commun. Statist. Theory Meth. 44(8) (2015), 1706-1734.
[14] M. Asadi, On the mean past lifetime of the components of a parallel system, J. Statist. Plann. Inference 136 (2006), 1197-1206.
[15] F. Ashkar and S. Mahdi, Fitting the log-logistic distribution by generalized moments, Journal of Hydrology 328 (2006), 694-703.
[16] I. Bairamov and M. Ozkut, Mean residual life and inactivity time of a coherent system subjected to Marshall-Olkin type shocks, J. Comput. Appl. Math. 298 (2016), 190-200.
[17] A. C. Bemmaor and N. Glady, Modeling purchasing behavior with sudden death: a flexible customer lifetime model, Management Science 58(5) (2012), 1012-1021.
[18] M. Bourguignon, R. B. Silva and G. M. Cordeiro, The Weibull-G family of probability distributions, Journal of Data Science 12 (2014), 53-68.
[19] H. Bozdogan, Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions, Psychometrika 52 (1987), 345-370.
[20] K. Brown and W. Forbes, A mathematical model of aging processes, Journal of Gerontology 29(1) (1974), 46-51.
[21] I. W. Burr, Cumulative frequency functions, Annals of Mathematical Statistics 13 (1942), 215-232.
[22] R. G. Clark, H. V. Henderson, G. K. Hoggard, R. S. Ellison and B. J. Young, The ability of biochemical and hematological tests to predict recovery in periparturient recumbent cows, NZ Veterinary Journal 35 (1987), 126-133.
[23] D. Collett, Modelling Survival Data in Medical Research, Chapman and Hall, London, 2003.
[24] D. Conen, V. Wietlisbach, P. Bovet, C. Shamlaye, W. Riesen, F. Paccaud and M. Burnier, Prevalence of hyperuricemia and relation of serum uric acid with cardiovascular risk factors in a developing country, BMC Public Health 4 (2004), Article number 9. http://www.biomedcentral.com/1471-2458/4/9.
[25] G. M. Cordeiro and M. de-Castro, A new family of generalized distributions, J. Stat. Comput. Simul. 81 (2011), 883-893.
[26] G. M. Cordeiro, E. M. M. Ortega and G. Silva, The beta extended Weibull family, Journal of Probability and Statistical Science 10 (2012), 15-40.
[27] G. M. Cordeiro, E. M. M. Ortega and D. C. C. da-Cunha, The exponentiated generalized class of distributions, Journal of Data Science 11 (2013), 1-27.
[28] G. M. Cordeiro, M. Alizadeh and E. M. M. Ortega, The exponentiated half-logistic family of distributions: properties and applications, J. Probab. Stat. (2014), Article ID 864396, 21 pp.
[29] F. Domma and F. Condino, The beta-Dagum distribution: definition and properties, Commun. Statist. Theory Meth. 44 (2013), 4070-4090.
[30] A. C. Economos, Rate of aging, rate of dying and the mechanism of mortality, Archives of Gerontology and Geriatrics 1(1) (1982), 46-51.
[31] P. Ein-Dor and J. Feldmesser, Attributes of the performance of central processing units: a relative performance prediction model, Comm. ACM 30 (1987), 308-317.
[32] N. Eugene, C. Lee and F. Famoye, Beta-normal distribution and its applications, Commun. Statist. Theory Meth. 31 (2002), 497-512.
[33] P. R. Fisk, The graduation of income distribution, Econometrica 29 (1961), 171-185.
[34] I. Ghosh and M. Bourguignon, A new extended Burr XII distribution, Austrian Journal of Statistics 46 (2017), 33-39.
[35] H. Goual and H. M. Yousof, Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test, J. Appl. Stat. 47(3) (2020), 393-423.
[36] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000.
[37] W. Gui, Marshall-Olkin extended log-logistic distribution and its application in minification processes, Appl. Math. Sci. 7(80) (2013), 3947-3961.
[38] Y. Guney and O. Arslan, Robust parameter estimation for the Marshall-Olkin extended Burr XII distribution, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 66(2) (2017), 141-161.
[39] E. J. Hannan and B. G. Quinn, The determination of the order of an autoregression, Journal of the Royal Statistical Society, Series B 41 (1979), 190-195.
[40] S. Heritier, E. Cantoni, S. Copt and M. Victoria-Feser, Robust Methods in Biostatistics, John Wiley & Sons, New York, 2009.
[41] F. Jamal, M. A. Nasir, M. H. Tahir and N. H. Montazeri, The odd Burr-III family of distributions, Journal of Statistics Applications and Probability 6 (2017), 105-122.
[42] N. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Volume 1, 2nd ed., Wiley, New York, 1994.
[43] N. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Volume 2, 2nd ed., Wiley, New York, 1995.
[44] M. C. Jones, Families of distributions arising from distributions of order statistics, Test 13 (2004), 1-43.
[45] J. Kenney and E. Keeping, Mathematics of Statistics, Volume 1, 3rd ed., Princeton, Van Nostrand, New Jersey, 1962.
[46] C. Kleiber and S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences, John Wiley, New York, 2003.
[47] J. D. Kloke and J. W. Mckean, R fit: rank-based estimation for linear models, The R Journal 4 (2012), 57-64.
[48] J. F. Lawless, Statistical Models and Methods for lifetime data, John Wiley, New York, 2003.
[49] A. J. Lemonte, The beta log-logistic distribution, Braz. J. Probab. Stat. 28(3) (2014), 313-332.
[50] S. R. Lima and G. M. Cordeiro, The extended log-logistic distribution: properties and application, Annals of the Brazilian Academy of Sciences 89(1) (2017), 3-17.
[51] M. E. Mead, A new generalization of Burr XII distribution, Journal of Statistics: Advances in Theory and Applications 12(2) (2014), 53-73.
[52] J. J. A. Moors, A quantile alternative for kurtosis, The Statistician 37 (1998), 25-32.
[53] H. M. Moustafa and S. G. Ramadan, Errors of misclassification and their probability distributions when the parent populations are Gompertz, Appl. Math. Comput. 163 (2005), 423-442.
[54] S. Nadarajah, G. M. Cordeiro and E. M. M. Ortega, The Zografos-Balakrishnan-G family of distributions: Mathematical properties and applications, Commun. Statist. Theory Meth. 44 (2015), 186-215.
[55] M. A. Nasir, M. H. Tahir, F. Jamal and G. Ozel, A new generalized Burr family of distributions for the lifetime data, Journal of Statistics Applications and Probability 6(2) (2017), 401-417.
[56] P. F. Paranaíba, E. M. M. Ortega, G. M. Cordeiro and R. R. Pescim, The beta Burr XII distribution with application to lifetime data, Comput. Statist. Data Anal. 55 (2011), 1118-1136.
[57] P. F. Paranaíba, E. M. M. Ortega, G. M. Cordeiro and M. A. de Pascoa, The Kumaraswamy Burr XII distribution: theory and practice, J. Stat. Comput. Simul. 83 (2013), 2117-2143.
[58] R. R. Pescim, G. M. Cordeiro, C. G. B. Demetrio, E. M. M. Ortega and S. Nadarajah, The new class of Kummer beta generalized distributions, Statistics and Operations Research Transactions (SORT) 36 (2012), 153-180.
[59] A. P. Prudnikov, Y. A. Brychkov and O. I. Morichev, Integrals and Series, Vol. 4: Direct Laplace Transforms, Gordon and Breach Science Publishers, New York, London, 1992.
[60] A. Rényi, On measures of entropy and information, 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 1961, pp. 547-561.
[61] S. Rezaei, B. B. Sadr, M. Alizadeh and S. Nadarajah, Topp-Leone generated family of distributions: properties and applications, Commun. Statist. Theory Meth. 46(6) (2017), 2893-2909.
[62] M. M. Ristiec and N. Balakrishnan, The gamma-exponentiated exponential distribution, J. Stat. Comput. Simul. 82 (2012), 1191-1206.
[63] T. V. F. de Santana, E. M. M. Ortega, G. M. Cordeiro and G. O. Silva, The Kumaraswamy-log-logistic distribution, J. Stat. Theory Appl. 11(3) (2012), 265-291.
[64] G. Schwarz, Estimating the dimension of a model, Ann. Statist. 6 (1978), 461-464.
[65] C. E. Shannon, Prediction and entropy of printed English, Bell System Technical Journal 30 (1951), 50-64.
[66] H. Shono, Efficiency of the finite correction of Akaike’s information criteria, Fisheries Science 66 (2000), 608-610.
[67] R. B. Silva and G. M. Cordeiro, The Burr XII power series distributions: a new compounding family, Braz. J. Probab. Stat. 29(3) (2015), 565-589.
[68] P. R. Tadikamalla, A look at the Burr and related distributions, International Statistical Review 48 (1980), 337-344.
[69] M. H. Tahir and S. Nadarajah, Parameter induction in continuous univariate distributions: well-established G families, Anada Academia Brasileira de Cincias (Annals of the Brazilian Academy of Sciences) 87 (2015), 539-568.
[70] M. H. Tahir, M. Mansoor, M. Zubair and G. G. Hamedani, McDonald log- logistic distribution, J. Stat. Theory Appl. 13 (2014), 65-82.
[71] M. H. Tahir, G. M. Cordeiro, M. Alizadeh, M. Mansoor, M. Zubair and G. G. Hamedani, The odd generalized exponential family of distributions with applications, Journal of Statistical Distributions and Applications 2(1) (2015), 1 28.
[72] M. H. Tahir, G. M. Cordeiro, A. Alzaatreh, M. Mansoor and M. Zubair, The logistic-X family of distributions and its applications, Commun. Statist. Theory Meth. 45 (2016a), 7326-7349.
[73] M. H. Tahir, M. Zubair, M. Mansoor, G. M. Cordeiro, M. Alizadeh and G. G. Hamedani, A new Weibull-G family of distributions, Hacet. J. Math. Stat. 45 (2016b), 629-647.
[74] H. Torabi and N. H. Montazeri, The gamma-uniform distribution and its application, Kybernetika 48 (2012), 16-30.
[75] H. Torabi and N. H. Montazeri, The logistic-uniform distribution and its applications, Commun. Statist. Simulation Comp. 43 (2014), 2551-2569.
[76] W. N. Venables and B. D. Ripley, Modern Applied Statistics with S, 4th ed., Springer, New York, 2002.
[77] P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Corresp. Math. Phys. 10 (1838), 113-121.
[78] S. Weisberg, Applied Linear Regression, 4th ed., Wiley, Hoboken, NJ, 2014. URL: http://z.umn.edu/alr4ed.
[79] W. Willemse and H. Kappelaar, Knowledge elicitation of Gompertz law of mortality, Scand. Actuar. J. 2 (2000), 168-179.
[80] K. Zografos and N. Balakrishnan, On families of beta- and generalized gamma-generated distributions and associated inference, Stat. Methodol. 6 (2009), 344-362.