Omid Kharazmi, Ali Saadatinik
2.003 591


A new class of distributions called the hyperbolic cosine – F (HCF) distribution is introduced and its properties are explored.This new class of distributions is obtained by compounding a baseline F distribution with the hyperbolic cosine function. This technique resulted in adding an extra parameter to a family of distributions for more flexibility. A special case with two parameters has been considered in details namely; hyperbolic cosine exponential (HCE) distribution. Various properties of the proposed distribution including explicit expressions for the moments, quantiles, moment generating function, failure rate function, mean residual lifetime, order statistics, stress-strength parameter and expression of the Shannon entropy are derived. Estimations of parameters in HCE distribution for two data sets obtained by eight estimation procedures: maximum likelihood, Bayesian, maximum product of spacings, parametric bootstrap, non-parametric bootstrap, percentile, least-squares and weighted least-squares. Finally these data sets have been analyzed for illustrative purposes and it is observed that in both cases the proposed model fits better than Weibull, gamma and generalized exponentialdistributions.


Hyperbolic cosine function, Exponential distribution, Mean residual life time, Maximum product of spacings, Maximum likelihood estimation, Bootstrap.

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Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36(1), 106-108.

Alexander, C., Cordeiro, G. M., Ortega, E. M., & Sarabia, J. M. (2012). Generalized beta-generated distributions. Computational Statistics & Data Analysis, 56(6), 1880-1897.

Alizadeh, M., Cordeiro, G. M., De Brito, E., & Demétrio, C. G. B. (2015a). The beta Marshall-Olkin family of distributions. Journal of Statistical Distributions and Applications, 2(1), 1.

Alizadeh, M., Emadi, M., Doostparast, M., Cordeiro, G. M., Ortega, E. M., & Pescim, R. R. (2015b). A new family of distributions: the Kumaraswamy odd log-logistic, properties and applications. Hacettepa Journal of Mathematics and Statistics, forthcomig.

Alizadeh, M., Tahir, M. H., Cordeiro, G. M., Mansoor, M., Zubair, M., & Hamedani, G. G. (2015c). The Kumaraswamy Marshal-Olkin family of distributions. Journal of the Egyptian Mathematical Society, 23(3), 546-557.

Alzaatreh, A., Lee, C., & Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71(1), 63-79.

Alzaghal, A., Famoye, F., & Lee, C. (2013). Exponentiated $ T $-$ X $ Family of Distributions with Some Applications. International Journal of Statistics and Probability, 2(3), 31.

Amini, M., MirMostafaee, S. M. T. K., & Ahmadi, J. (2014). Log-gamma-generated families of distributions. Statistics, 48(4), 913-932.

Andrews, D. F., & Herzberg, A. M. (1985). Prognostic variables for survival in a randomized comparison of treatments for prostatic cancer. In Data (pp. 261-274). Springer New York.

Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171-178.

Barlow, R. E., Toland, R. H., & Freeman, T. (1984). A Bayesian analysis of stress-rupture life of kevlar 49/epoxy spherical pressure vessels. In Proc. Conference on Applications of Statistics’, Marcel Dekker, New York.

Bourguignon, M., Silva, R. B., & Cordeiro, G. M. (2014). The Weibull-G family of probability distributions. Journal of Data Science, 12(1), 53-68.

Cheng, R. C. H., & Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society. Series B (Methodological), 394-403.

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of statistical computation and simulation, 81(7), 883-898.

Cordeiro, G. M., Alizadeh, M., & Diniz Marinho, P. R. (2016). The type I half-logistic family of distributions. Journal of Statistical Computation and Simulation, 86(4), 707-728.

Cordeiro, G. M., Alizadeh, M., & Ortega, E. M. (2014a). The exponentiated half-logistic family of distributions: Properties and applications. Journal of Probability and Statistics, 2014.

Cordeiro, G. M., Ortega, E. M., & da Cunha, D. C. (2013). The exponentiated generalized class of distributions. Journal of Data Science, 11(1), 1-27.

Efron, B., & Tibshirani, R. J. (1994). An introduction to the bootstrap. CRC press.

Eugene, N., Lee, C., & Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and methods, 31(4), 497-512.

Glaser, R. E. (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association, 75(371), 667-672.

Gupta, R. C., Gupta, P. L., & Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory and methods, 27(4), 887-904.

Gupta, R. D., & Kundu, D. (2009). A new class of weighted exponential distributions. Statistics, 43(6), 621-634.

Jones, M. C. (2004). Families of distributions arising from distributions of order statistics. Test, 13(1), 1-43.

Kao, J. H. (1958). Computer methods for estimating Weibull parameters in reliability studies. IRE Transactions on Reliability and Quality Control, 15-22.

Kao, J. H. (1959). A graphical estimation of mixed Weibull parameters in life-testing of electron tubes. Technometrics, 1(4), 389-407.

Kharazmi, O., Mahdavi, A., & Fathizadeh, M. (2015). Generalized Weighted Exponential Distribution. Communications in Statistics-Simulation and Computation, 44(6), 1557-1569.

Marshall, A. W., & Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84(3), 641-652.

Merovci, F., Alizadeh, M., & Hamedani, G. G. (2016). Another generalized transmuted family of distributions: properties and applications. Austrian Journal of Statistics, 45, 71-93.

Murthy, D. P., Xie, M., & Jiang, R. (2004). Weibull models (Vol. 505). John Wiley & Sons.

Nadarajah, S., Cancho, V. G., & Ortega, E. M. (2013a). The geometric exponential Poisson distribution. Statistical Methods & Applications, 22(3), 355-380.

Nadarajah, S., Nassiri, V., & Mohammadpour, A. (2014). Truncated-exponential skew-symmetric distributions. Statistics, 48(4), 872-895.

Nelder, J. A., & Mead, R. (1965). A simplex method for function minimization. The computer journal, 7(4), 308-313.

Ramos, M. W., Marinho, P. R., Silva, R. V., & Cordeiro, G. M. (2013). The exponentiated Lomax Poisson distribution with an application to lifetime data. Advances and Applications in Statistics, 34(2), 107-135.

Ranneby, B. (1984). The maximum spacing method. An estimation method related to the maximum likelihood method. Scandinavian Journal of Statistics, 93-112.

R Development, C. O. R. E. TEAM 2011: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.

Ristić, M. M., & Balakrishnan, N. (2012). The gamma-exponentiated exponential distribution. Journal of Statistical Computation and Simulation, 82(8), 1191-1206.

Shannon, C. E. (1948). A mathematical theory of communication, bell System technical Journal 27: 379-423 and 623–656. Mathematical Reviews (MathSciNet): MR10, 133e.

Swain, J. J., Venkatraman, S., & Wilson, J. R. (1988). Least-squares estimation of distribution functions in Johnson's translation system. Journal of

Statistical Computation and Simulation, 29(4), 271-297.

Tahir, M. H., Cordeiro, G. M., Alzaatreh, A., Mansoor, M., & Zubair, M. (2016). The Logistic-X family of distributions and its applications. Communications in Statistics-Theory and Methods, (just-accepted).

Tahir, M. H., Cordeiro, G. M., Alizadeh, M., Mansoor, M., Zubair, M., & Hamedani, G. G. (2015b). The odd generalized exponential family of distributions with applications. Journal of Statistical Distributions and Applications, 2(1), 1.

Torabi, H., & Hedesh, N. M. (2012). The gamma-uniform distribution and its applications. Kybernetika, 48(1), 16-30.

Torabi, H., & Montazeri, N. H. (2014). The Logistic-Uniform Distribution and Its Applications. Communications in Statistics-Simulation and Computation, 43(10), 2551-2569.

Zografos, K., & Balakrishnan, N. (2009). On families of beta-and generalized gamma-generated distributions and associated inference. Statistical Methodology, 6(4), 344-362.