A weighted Topp-Leone G family of distributions: properties, applications for modelling reliability data and different method of estimation

Based on the Topp-Leone distribution, we propose a new family of continuous distributions with one shape parameter called the weighted Topp-Leone family. We study some basic properties including quantile function, asymptotic, mixture for cdf and pdf, various entropies and order statistics.Then we study Lindley case as special case with more details. The maximum likelihood estimates of parameters are compared with various methods of estimations by conducting a simulation study. Finally, three real data sets are illustration the purposes.

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  • [1] M. Alizadeh, F. Lak, M. Rasekhi, T.G. Ramires, H.M. Yousof and E. Altun, The odd log-logistic ToppLeone G family of distributions: heteroscedastic regression models and applications, Comput. Stat. 33 (3), 1217-1244, 2018.
  • [2] A. Al-Shomrani, O. Arif, A. Shawky, S. Hanif and M.Q. Shahbaz, Topp-Leone family of distributions: some properties and application, Pak. J. Stat. Oper. Res. 12 (3), 443-451, 2016.
  • [3] A. Alzaatreh, C. Lee and F. Famoye, A new method for generating families of continuous distributions, Metron 71 (1), 63-79, 2013.
  • [4] T.W. Anderson and D.A. Darling, Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes, Ann. Math. Stat., 193-212, 1952.
  • [5] N. Balakrishnan, Order statistics from the half logistic distribution, J. Stat. Comput. Simul. 20 (4), 287-309, 1985.
  • [6] R.A. Bantan, F. Jamal, C. Chesneau and M. Elgarhy, A new power ToppLeone generated family of distributions with applications, Entropy 21 (12) 1177, 2019.
  • [7] L. Benkhelifa, The MarshallOlkin extended generalized Lindley distribution: Properties and applications, Comm. Statist. Simulation Comput. 46 (10), 8306-8330, 2017.
  • [8] E. Brito, G.M. Cordeiro, H.M. Yousof, M. Alizadeh and G.O. Silva, The ToppLeone odd log-logistic family of distributions, J. Stat. Comput. Simul. 87 (15), 3040-3058, 2017.
  • [9] K. Choi and W. Bulgren, An estimation procedure for mixtures of distributions, J. R. Stat. Soc. Ser. B. Stat. Methodol., 444-460, 1968.
  • [10] G.M. Cordeiro and M. de Castro, A new family of generalized distributions, J. Stat. Comput. Simul. 81 (7), 883-898, 2011.
  • [11] R.M. Corless, G.H. Gonnet, D.E. Hare, D.J. Jeffrey and D.E. Knuth, On the LambertW function, Adv Comput Math 5 (1), 329-359, 1996.
  • [12] S. Dey, J. Mazucheli and S. Nadarajah, Kumaraswamy distribution: different methods of estimation, Comput. Appl. Math., 1-18, 2017.
  • [13] M. Elgarhy, M. Arslan Nasir, F. Jamal and G. Ozel, The type II Topp-Leone generated family of distributions: Properties and applications, Int. j. stat. manag. syst. 21 (8), 1529- 1551, 2018.
  • [14] M.E. Ghitany, D.K. Al-Mutairi, N. Balakrishnan and L.J. Al-Enezi, Power Lindley distribution and associated inference, Comput Stat Data Anal 64, 20-33, 2013.
  • [15] M.E. Ghitany, B. Atieh and S. Nadarajah, Lindley distribution and its application, Math Comput Simul 78 (4), 493-506, 2008.
  • [16] J.U Gleaton and J.D. Lynch, Properties of generalized log-logistic families of lifetime distributions, J Probab Stat 4 (1), 51-64, 2006.
  • [17] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 7 edn, Academic Press, New York, 2007.
  • [18] R.D. Gupta and D. Kundu, A new class of weighted exponential distributions, Statistics 43 (6), 621-634, 2009.
  • [19] R.D. Gupta and D. Kundu, Theory and methods: Generalized exponential distributions, Aust N Z J Stat 41 (2), 173-188, 1999.
  • [20] A.S. Hassan, M. Elgarhy and Z. Ahmad, Type II generalized topp-Leone family of distributions: properties and applications, Data Sci. J. 17 (4), 2019.
  • [21] M.C. Jones, Families of distributions arising from distributions of order statistics, Test 13 (1), 1-43, 2004.
  • [22] M.Ç. Korkmaz, H.M. Yousof, M. Alizadeh and G.G. Hamedani, The Topp-Leone generalized odd log-logistic family of distributions: properties, characterizations and applications, Commun. Fac. Sci 68 (2), 1506-1527, 2019.
  • [23] M.R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer Science and Business Media, 2012.
  • [24] D.P. Murthy, M. Xie, and R. Jiang, Weibull Models, John Wiley and Sons, 2004.
  • [25] S. Nadarajah, H.S. Bakouch and R. Tahmasbi, A generalized Lindley distribution, Sankhya B 73 (2), 331-359, 2011.
  • [26] G. Ozel, M. Alizadeh, S. Cakmakyapan, G.G. Hamedani, E.M. Ortega and V.G. Cancho, The odd log-logistic Lindley Poisson model for lifetime data, Comm. Statist. Simulation Comput. 46 (8), 6513-6537, 2017.
  • [27] V. Ranjbar, M. Alizadeh and E. Altun, Extended generalized Lindley distribution: properties and applications, J. Math. Ext. 13, 117-142, 2019.
  • [28] A. Rényi, On measures of entropy and information in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1, 547-561, 1961.
  • [29] H. Reyad, M.Ç. Korkmaz, A.Z. Afify, G.G. Hamedani and S. Othman, The Fréchet Topp Leone-G family of distributions: Properties, characterizations and applications, Ann. Data Sci. 8 (2), 345-366, 2021.
  • [30] H. Reyad, M. Alizadeh, F. Jamal and S. Othman, The Topp Leone odd Lindley-G family of distributions: properties and applications, Int. j. stat. manag. syst. 21 (7), 1273-1297, 2018.
  • [31] S. Rezaei, B.B. Sadr, M. Alizadeh and S. Nadarajah, Topp-Leone generated family of distributions: Properties and applications, Comm. Statist. Theory Methods 46 (6), 2893-2909, 2017.
  • [32] Y. Sangsanit and W. Bodhisuwan, The Topp-Leone generator of distributions: properties and inferences, Songklanakarin Journal of Science and Technology 38 (5), 2016.
  • [33] C.E. Shannon, Prediction and entropy of printed English, Bell Labs Technical Journal 30, 50-64, 1951.
  • [34] J.J. Swain, S. Venkatraman and J.R. Wilson, Least-squares estimation of distribution functions in Johnson’s translation system, JJ. Stat. Comput. Simul. 29, 271- 297, 1988.
  • [35] C.W. Topp and F.C. Leone, A family of J-shaped frequency functions, J. Amer. Statist. Assoc. 50 (269), 209-219, 1955.
  • [36] H.M. Yousof, M. Alizadeh, S.M.A. Jahanshahi, T.G. Ramires, I. Ghosh and G.G. Hamedani, The transmuted Topp-Leone G family of distributions: theory, characterizations and applications, Data Sci. J. 15 (4), 723-740, 2017.
  • [37] H.M. Yousof and M.Ç Korkmaz, Topp-Leone Nadarajah-Haghighi distribution, İstatistikçiler Dergisi: İstatistik ve Aktüerya 10 (2), 119-127, 2017.