A new generalized Logistic class of distributions: Properties and applications on flood and earthquake data sets with bivariate extension

Sadaf Khan; Farrukh Jamal; Muhammad Hussain Tahir; Ahmed Elhassanein

doi:10.64060/JASR.v1.i2.0

Authors

Sadaf Khan Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur-63100, Pakistan Author
Farrukh Jamal Department of Statistics , Faculty of Science, University of Tabuk, Tabuk , Saudi Arabia Author
Muhammad Hussain Tahir Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur-63100, Pakistan Author
Ahmed Elhassanein Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia Author

DOI:

https://doi.org/10.64060/JASR.v1.i2.0

Keywords:

Generalized Family, Logistic-G Family, Burr-III Distribution, Maximum Likelihood Method, Bi-variate Modelling, Surface Plots

Abstract

For univariate and bi-variate data, we propose a new generalized logistic class of distributions exible enough to exhibit monotone and non-monotone hazard rates shapes. The physical interpretation of the new family preludes in the context of series-parallel structures. Its mathematical features, including a valuable expansion for the density, explicit formulations for the quantile function, ordinary and incomplete moments, and generating function, are all derived. The parameter estimation of the new family is done using the maximum likelihood method. One of the unique model, called the generalized logistic Burr-III, is thoroughly investigated in applied sense. The exible density and hazard rate shapes capacitates the model to be applicable in extreme value theory. For univariate case, two real-life data sets related to hydrology and seismic activity have been employed to solidify the superiority of the proposed distribution to ve well established families. For bivariate data, initially a bivariate extension of the proposed family is established analytically with the help of empirical ndings. Then, a real bi-variate data related to operational lifespan of two components of a computer, has been studied using bivariate generalized logistic Burr-III model and the results are reported.

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References

[1] Azzalini. A, A class of distributions which includes the normal ones, Scandanavian Journal of Statistics, 12 (1985), pp. 171{178.

[2] A.W. Marshall and I. Olkin, A new method for adding parameters to a family of distributions with application to the exponential and Weibull families, Biometrika 84 (1997), pp. 641{652.

[3] R.C. Gupta, P.L. Gupta and R.D. Gupta, Modeling failure time data by Lehman alternatives, Commun. Stat. Theory Methods 27 (1998), pp. 887{904.

[4] J.U. Gleaton and J.D. Lynch, Properties of generalized log-logistic families of lifetime distributions J. Probab. Stat. Sci. 4 (2006), pp. 51{64.

[5] W.T. Shaw and I.R. Buckley, The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map, (2007), UCL discovery repository, http://discovery.ucl.ac.uk/id/eprint/643923.

[6] K. Zografos and N. Balakrishnan, On families of beta- and generalized gamma generated distributions and associated inference, Stat. Methodol. 6 (2009), pp. 344{362.

[7] M.M. Ristic and N. Balakrishnan, The gamma-exponentiated exponential distribution, J.Stat. Comput. Simul. 82 (2012), pp. 1191{1206.

[8] M. Bourguignon, R.B. Silva and G.M. Cordeiro, The Weibull-G family of probability distributions, J. Data. Sci. 12 (2014), pp. 53{68.

[9] N. Eugene, C. Lee and F. Famoye, Beta-normal distribution and its applications, Commun. Stat. Theory Methods 31 (2002), pp. 497{512.

[10] G.M. Cordeiro and M. de-Castro, A new family of generalized distributions J. Stat. Comput. Simul. 81 (2011), pp. 883{898.

[11] G.M. Cordeiro, E.M.M. Ortega and D.C.C. Cunha, The exponentiated generalized class of distributions J. Data. Sci. 11 (2013), pp. 1{27.

[12] C. Alexander, G.M. Cordeiro, E.M.M. Ortega and J.M. Sarabia, Generalized beta-generated distributions Comput. Stat. Data Anal. 56 (2012), pp. 1880{1897.

[13] M.H. Tahir and S. Nadarajah, Parameter induction in continuous univariate distributions: Wellestablished G families, An. Acad. Bras. Ci^enc. 87 (2015), pp. 539{568.

[14] M.H. Tahir and G.M. Cordeiro, Compounding of distributions: a survey and new generalized classes, J. Stat. Dist. Applic. 3 (2016), Art. 13, doi:10.1186/s40488-016-0052-1.

[15] M. Alizadeh, M. C. Korkamaz, J. A. Almamy and A. A. E. Ahmed, Another odd log-logistic logarithmic class of continuous distributions. Istatistikciler Dergisi: Istatistik ve Aktuerya, 2018, 11.2:55-72.

[16] M. C. Korkamaz, M. Alizadeh, H.M. Yousof, and G.G. Hamedani, The Topp-Leone generalized oddlog-logistic family od distributions: Properties, characterizations and applications. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics (2019), 69(2): 1303-5991.

[17] G.M. Cordeiro, E. Altun, M. C. Korkamaz, R. Pescim, A.Z. Afy and H.M. Yousof, The xgamma Family: Censored Regression Modelling and Applications. REVSTAT (2020), 18(5): 593612.

[18] G.G. Hamedani, M. C. Korkamaz; N.S. Butt and H.M. Yousof, The Type I Quasi Lambert Family: Properties, Characterizations and Dierent Estimation Methods. Mathematical and Statistical Science Faculty Research and Publications (2021), 106. https://epublications.marquette.edu/mathfac/106

[19] E. Altun, M. C. Korkamaz, M. El-Morshedy and M.S. Eliwa, A New Flexible Family of Continuous Distributions: The Additive Odd-G Family. Mathematics. 2021; 9(16):1837. https://doi.org/10.3390/math9161837

[20] Q. Ramzan, M. Amin, A. Elhassanein and M. Ikram, The extended generalized inverted Kumaraswamy Weibull distribution: Properties and applications, AIMS Mathematics 6 (9) (2021),pp. 9955-9980.

[21] Q. Ramzan, et al., On the extended generalized inverted Kumaraswamy distribution, Computational Intelligence and Neuroscience 2022 (2022), 1612959.

[22] R. A. R. Bantan et al., Statistical analysis of COVID-19 data: using a new univariate and bivariate statistical model, Journal of Function Spaces 2022 (2022), 2851352.

[23] A. Alzaatreh, F. Famoye and C. Lee, A new method for generating families of continuous distributions, Metron 71 (2013), pp. 63{79.

[24] H. Torabi and N.H. Montazari, The logistic-uniform distribution and its application, Commun. Stat.Simul. Comput. 43 (2014), pp. 2551{2569.

[25] M.H. Tahir, G.M. Cordeiro, M. Alizadeh, M. Mansoor and M. Zubair, The logistic-X family of distributions and its applications, Commun. Stat. Theory Methods 45 (2016), pp. 7326{7349.

[26] M. Mansoor, M.H. Tahir, G.M. Cordeiro, E.M.M. Ortega, A. Alzaatreh, On analyzing non-monotone failure data. To appear in REVSTAT Journal, 2021.

[27] S. Nasiru, P. N. Mwitaand O. Ngesa, Exponentiated generalized transformed-transformer family of distributions, J. of Statistical and Econometric Methods, 6(4) 2017,pp. 1{17.

[28] Alzaghal, A., Famoye, F. and Lee, C.(2013). Exponentiated T-X family of distributions with some applications. International Journal of Probability and Statistics 2, 31{49.

[29] G.S. Mudholker, D.K. Srivastava, Exponentiated Weibull family for analyzing bathtub failure data, IEEE Transactions of Reliability 42 (1993), pp. 299{302.

[30] R.D. Gupta and D. Kundu, Exponentiated exponential family: An alternative to gamma and Weibull distributions, Biometrical Journal 42 (2001), pp. 117{130.

[31] S. Nadarajah and S. Kotz, The exponentiated type distributions, Acta Applicandae Mathematica, 92 (2006a), pp.97{111.

[32] Apostol, T. M. (1974), Mathematical analysis, Addison-Wesley Pub. Co.

[33] Kleiber, C. and Kotz, S., Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience (2003); Application-stock returns.

[34] F. Domma and F. Condino, The Beta-Dagum Distribution: Denition and Properties, Communications in Statistics - Theory and Methods, 42:22 (2013), pp. 4070{4090.

[35] G.M. Cordeiro, E.M.M. Ortega, and S. Nadarajah, The Kumaraswamy Weibull distribution with application to failure data, J. Franklin Inst. 347 (2010), pp. 1399{1429.

[36] B.O. Oluyede, S. Huang and M. Pararai, A new class of generalized Dagum distribution with application to income and lifetime data, J. of Stat. and Econometric methods, 3 (2014), pp. 125{151.

[37] A. Y. Al-Saiari, S. A. Mousa and L. A. Baharith, Marshall-Olkin extended Burr III distribution, International Mathematical Forum, 11 (2016), pp. 631{642.

[38] A. Asgharzadeh, H.S. Bakouch and M. Habibi, A generalized binomial exponential 2 distribution:modeling and applications to hydrologic events, J. Appl. Statist. 44 (2017), pp.2368{2387.

[39] Kus, C., A new lifetime distribution, Comput. Statist. Data Anal. 51 (2007), pp. 4497-4509., pp. 79{88.

[40] J. A.Darwish, L. I. Al turk, and M. Q. Shahbaz, Bivariate transmuted Burr distribution: Properties and applcations, Pak.j.stat.oper.res., 17 (1) (2021), pp. 15-24.

[41] M.Ganji, H. Bevraninand, and H. Golzar, A new method for generating a continuous bivariate distribution families, JIRSS, 17(1) (2018), pp. 109-129.

[42] H. Z. Muhammed, On a bivariate generalized inverted Kumaraswamy distribution, Physica A: Statistical Mechanics and its Applications, 553 (2020), article 124281.

[43] S. Mondal and D. Kundu, A bivariate inverse Weibull distribution and its applications in complementary risks models, Journal of Applied Statistics, 47 (6) (2021), pp. 1084{1108,

[44] J. A. Darwish and M. Q. Shahbaz, Bivariate transmuted Burr distribution: properties and applcations, Pakistan Journal of Statistics & Operation Research, 17 (1) (2021), pp. 15{24.

36[45] M. V. O.Peres, J. A.Achcar and E. Z. Martinez, Bivariate lifetime models in presence of cure fraction:a comparative study with many dierent copula functions. Heliyon, 6 (2020), e03961.

[46] A. Elhassanein, On statistical properties of a ewn bivariate modied Lindley distribution with anapplication to nancial data. Complexity, 2022 (2022), 2328831.

[47] A. Elhassanein, A new type I half-logistic inverse Weibull distribution with an application to the relief times data of patients receiving an analgesic. Journal of Medical Imaging and Health Informatics 12(2) (2022), pp. 184-199.

[48] J.Navarro, Characterizations using the bivariate failure rate function. Statistics and Probability Letters, 78(12) (2010), p. 1349.

[49] T. S. Durrani and X. Zeng. (2007) Copulas for bivariate probability distributions," Electronics Letters, 43 (4) (2007).

[50] R. P. Oliveira, J. A. Achcar, J. Mazucheli, and W. Bertoli, A new class of bivariate Lindley distributions based on stress and shock models and some of their reliability properties,"Reliability Engineering and System Safety, 211 (2021), article 107528.

[51] M. Ragab and A. Elhassanein, A new bivariate extended generalized inverted Kumaraswamy Weibull distribution. Advances in Mathematical Physics, 2022 (2022), 1243018.