# The Zografos-Balakrishnan Odd Log-Logistic Generalized Half-Normal Distribution with Mathematical Properties and Simulations

### Abstract

In this paper, A new class of distributions called the Zografos-Balakrishnan odd log-logistic Generalized half-normal (ZOLL-GHN) family with four parameters is introduced and studied. Useful representations and some mathematical properties of the new family include moments, quantile function, moment Generating function are investigated. The maximum likelihood equations for estimating the parameters based on real data are given. Different methods have been used to estimate its parameters such as maximum likelihood, Least squares, weighted least squares, Crammer-von-Misers,Anderson-Darling and right-tailed Anderson-Darling methods. We assesses the performance of the maximum likelihood estimators in terms of biases and mean squared errors by means of a simulation study. Finally, the usefulness of the family and fitness capability of this model, are illustrated by means of two real data sets.### References

K. Aas, and I. Haff, The generalized hyperbolic skew student’s t-distribution, Journal of Financial Econometrics, vol. 4, pp. 275–309, 2006.

C. Alexander, G. M. Cordeiro, E. M. M. Ortega, and J. M. Sarabia, Generalized beta-Generated distributions , Computational Statistics and Data Analysis, vol. 56, pp. 1880–1897, 2012.

M. Alizadeh, M. Emadi, M. Doostparast, G. M. Cordeiro, E. M. M. Ortega, and R. R. Pescim, A new family of distributions: the Kumaraswamy odd log-logistic, properties and applications , Hacettepe Journal of Mathematics and Statistics, vol. 44, pp. 1491–1512, 2015.

T. W. Anderson, and D. A. Darling,Asymptotic theory of certain” goodness of fit” criteria based on stochastic processes , The annals of mathematical statistics, pp. 193–212, 1952.

K. Choi, and W. Bulgren, An estimation procedure for mixtures of distributions , Journal of the Royal Statistical Society. Series B (Methodological), pp. 444–460, 1968.

K. Cooray, and M. M. Ananda, A generalization of the half-normal distribution with applications to lifetime data , Communications in Statistics-Theory and Methods, vol. 37(9), pp. 1323–1337, 2008.

G. M. Cordeiro, and M. de Castro, A new family of Generalized distributions, Journal of Statistical Computation and Simulation,vol. 81, pp. 883–898, 2011.

G. M. Cordeiro, R. R. Pescim, and E. M. Ortega, The Kumaraswamy generalized half-normal distribution for skewed positive data,Journal of Data Science, vol. 10(2), pp. 195–224, 2012.

G. M. Cordeiro, S. Nadarajah, and E. M. M. Ortega, General results for the beta Weibull distribution, Journal of Statistical Computation and Simulation, vol. 83, pp. 1082–1114, 2013.

G.M.Cordeiro, M.Alizadeh,E.M.Ortega,and L.H.V.Serrano, The Zografos-Balakrishnan odd log-logistic family of distributions: Properties and applications , Hacettepe Journal of Mathematics and Statistics, vol. 45(6), pp. 1781–1803, 2016.

G. M. Cordeiro, M. Alizadeh, R. R. Pescim, and E. M. Ortega, The odd log-logistic generalized half-normal lifetime distribution: Properties and applications, Communications in Statistics-Theory and Methods, vol. 46(9), pp. 4195–4214, 2017a.

G. M. Cordeiro, M. Alizadeh, G. Ozel, B. Hosseini, E. M. M. Ortega, and E. Altun,The generalized odd log-logistic family of distributions: properties, regression models and applications, Journal of Statistical Computation and Simulation, vol. 87(5), pp.908–932, 2017b.

S. Dey, J. Mazucheli and S. Nadarajah, Kumaraswamy distribution: different methods of estimation, Computational and Applied Mathematics, pp. 1–18, 2017.

N. Eugene, C. Lee, and F. Famoye,Beta-normal distribution and its applications , Communications in Statistics-Theory and Methods, vol. 31, pp. 497–512, 2002.

H. Exton, Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs, Academic Press, New York, 1978.

J. U. Gleaton, and J. D. Lynch, Properties of generalized log logistic families of lifetime distributions , Journal of Probability and Statistical Science, vol. 4(1), pp. 51–64, 2006.

I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products, sixth edition, Academic Press, San Diego, 2000.

B. E. Hansen, Autoregressive conditional density estimation , International Economic Review, vol. 35, pp. 705–730, 1994. (1994).

N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, Volume 1, 2nd edition. John Wiley and Sons, New York, 1994.

N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, Volume 2, 2nd edition. John Wiley and Sons, NewYork, 1995.

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.

S. Nadarajah, G. M. Cordeiro and E. M. Ortega, The Zografos Balakrishnan-G family of distributions, Mathematical properties and applications , Communications in Statistics-Theory and Methods, vol. 44(1), pp. 186-215, 2015.

R. R. Pescim, C. G. Demtrio, G. M. Cordeiro, E. M. Ortega, and M. R. Urbano,The beta generalized half-normal distribution, Computational statistics and data analysis, vol. 54(4), pp. 945–957, 2010.

M. M. Ristic, and N. Balakrishnan, The gamma exponentiated exponential distribution , Journal of Statistical Computation and Simulation, pp. 1191–1206, 2012.

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.

J. J. Swain, S. Venkatraman, and J. R. Wilson, Least-squares estimation of distribution functions in johnson’s translation system , Journal of Statistical Computation and Simulation, vol. 29(4), pp. 271–297, 1988.

M. Trott, The Mathematica guidebook for numerics, Springer Science and Business Media, 2009.

K. Zografos, and N. Balakrishnan, on families of beta- and generalized gamma-generated distributions and associated inference,Statistical Methodology, vol. 6, pp. 344–362, 2009. (2009).

*Statistics, Optimization & Information Computing*,

*7*(1), 211-234. https://doi.org/10.19139/soic.v7i1.649

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