The Topp-Leone odd log-logistic Gumbel Distribution: Properties and Applications
AbstractIn this article, the Topp-Leone odd log-logistic Gumbel (TLOLL-Gumbel) family of distribution have beenstudied. This family, contains the very flexible skewed density function. We study many aspects of the new model like hazard rate function, asymptotics, useful expansions, moments, generating Function, R´enyi entropy and order statistics. We discuss maximum likelihood estimation of the model parameters. Further, we study flexibility of the proposed family are illustrated of two real data sets.
T. Andrade, H. Rodrigues, M. Bourguignon, and G. M. Cordeiro, The Exponentiated Generalized Gumbel Distribution, Revista Colombiana de Estadstica, vol. 38(1), pp. 123–143, 2015.
J. Beirlant, Y. Goegebeur, J. Segers, and J. Teugels, D. DeWaal, and C. Ferro, Statistics of Extremes: Theory and Applications, West Sussex, England: John Wiley and Sons Ltd, 2006.
W. Barreto-Souza, A. H. S. Santos, and G. M. Cordeiro, The beta generalized exponential distribution, Journal of Statistical Computation and Simulation, vol. 80(2), pp. 159–172, 2010.
E. Brito, G. O. Silva, G. M. Cordeiro, and C. G. B. Demtrio, The McDonald Gumbel model, Communications in Statistics - Theory and Methods, vol. 45(11), pp. 3367–3382, 2016.
E. Brito, G. M. Cordeiro, H. M. Yousof, M. Alizadeh, and G. O. Silva, The ToppLeone odd log-logistic family of distributions, Journal of Statistical Computation and Simulation, vol. 87(15), pp. 3040–3058, 2017.
G. M. Cordeiro, S. Nadarajah, and E. M. M. Ortega, The Kumaraswamy Gumbel distribution, Statistical Methods and Applications, vol. 21, pp. 139–168, 2012.
A. I. Genec, Moments of order statistics of Topp-Leone distribution, Statistical Papers, vol. 53(1), pp. 117–131, 2012.
A. I. Genec, Estimation of P(X > Y ) with Topp-Leone distribution., Journal of Statistical Computation and Simulation, vol. 83(2), pp. 326–339, 2013.
M. E. Ghitany, S. Kotz, and M. Xie, On some reliability measures and their stochastic orderings for the Topp-Leone distribution, Journal of Applied Statistics, vol. 32(7), pp. 715–722, 2005.
E. J. Gumbel, Statistics of Extremes, New York: Columbia University Press, 1958.
I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, Seventh ed. San Diego: Academic Press, 2007.
J. Gupta, M. Garg, and M. Gupta, The Lomax-Gumbel Distribution, Palestine Journal of Mathematics, vol. 5(1), pp. 35–42, 2016.
N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Vol. 2 (2nd ed.). New York: John Wiley and Sons, Inc, 1995.
K. K. Jose, Marshall-Olkin Family of Distributions and their applications in reliability theory, time series modeling and stressstrength analysis, Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS005), pp. 3918–3923, 2011.
H. Karamikabir, M. Afshari, M. Alizadeh, and G. G. Hamedani, A new extended generalized Gompertz distribution with statistical properties and simulations, Communications in Statistics - Theory and Methods, Published online, 2019.
S. Kotz, and S. Nadarajah, Extreme Value Distributions, London, Imperial College Press, 2000.
S. Kotz, and E. Seier, Kurtosis of the Topp-Leone distributions, InterStat, pp. 1–15, 2007.
D. Kundu, N. Kannan, and N. Balakrishnan, On the hazard function of Birnbaum-Saunders distribution and associated inference, Computational Statistics and Data Analysis, vol. 52, pp. 2692–2702, 2008.
V. Leiva, M. Barros, and G. A. Paula, Generalized Birnbaum-Saunders Models Using R. XI Escola de Modelos de Regressao, Recife, Brazil, 2009.
S. Nadarajah, and S. Kotz, Moments of some J-shaped distributions, Journal of Applied Statistics, vol. 30(3), pp. 311-317, 2003.
S. Nadarajah, and S. Kotz, The beta Gumbel distribution, Mathematical Problems in Engineering, vol. 2004(4), pp. 323–332, 2004.
S. Nadarajah, The exponentiated Gumbel distribution with climate application, Environmetrics, vol. 17, 13–23, 2006.
S. Nadarajah, and S. Kotz, The Exponentiated Type Distributions, Acta Applicandae Mathematicae, vol. 92, pp. 97–111, 2006.
S. Nadarajah, G. M. Cordeiro, and E. M. M. Ortega, The Zografos Balakrishnan-G Family of Distributions: Mathematical Properties and Applications, Communications in Statistics - Theory and Methods, vol. 44, 186–215, 2014.
J. Ownuk, The Beta Exponentiated Gumbel Distribution, Journal of The Iranian Statistical Society, vol. 14(2), pp. 1–14, 2015.
D. Ruppert, M. P. Wand, and R. J. Carroll, Semiparametric regression, Cambridge university press, Number 12, 2003.
T. N. Sindhu, M. Saleem, and M. Aslam, Bayesian estimation for Topp-Leone distribution under trimmed samples, Journal of Basic and Applied Scientific Research, vol. 3, pp. 347–360, 2013.
R. L. Smith, and J. C. Naylor, A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, Journal of Applied Statistics, vol. 36, pp. 358–369, 1987.
C.W. Topp, and F. C. Leone, A family of J-shaped frequency functions, Journal of the American Statistical Association, vol. 50(269), pp. 209–219, 1955.
A. A. Zghoul Record values from a family of J-shaped distributions, Statistica, vol. 71(3):355, 2011.
M. Zhou, D. W. Yang, Y. Wang, and S. Nadarajah, Some J-shaped distributions: sums, products and ratios. In: Reliability and Maintainability Symposium. RAMS’06., Annual; IEEE, pp. 175–181, 2006.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).