Hyperbolic Cosine–Exponentiated Exponential Lifetime Distribution and its Application in Reliability

Document Type: Research Paper

Author

Department of Statistics, Faculty of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

Abstract

Recently, Kharazmi and Saadatinik (2016) introduced a new family of lifetime distributions called hyperbolic cosine – F (HCF) distribution. In the present paper, it is focused on a special case of HCF family with exponentiated exponential distribution as a baseline distribution (HCEE). Various properties of the proposed distribution including explicit expressions for the moments, quantiles, mode, moment generating function, failure rate function, mean residual lifetime, order statistics and expression of the entropy are derived. Estimating parameters of HCEE distribution are obtained by eight estimation methods: maximum likelihood, Bayesian, maximum product of spacings, parametric bootstrap, non-parametric bootstrap, percentile, least-squares and weighted least-squares. A simulation study is conducted to examine the bias, mean square error of the maximum likelihood estimators. Finally, one real data set has been analyzed for illustrative purposes and it is observed that the proposed model fits better than Weibull, gamma and generalized exponential distributions.

Keywords

Main Subjects


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

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

Alizadeh, M., Emadi, M., Doostparast, M., Cordeiro, G. M., Ortega, E. M., and 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., and Hamedani, G. G. (2015c). The Kumaraswamy Marshal-Olkin family of distributions. Journal of the Egyptian Mathematical Society, Vol. 23(3), pp. 546-557.

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

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

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

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

Azzalini, A. (2013). The skew-normal and related families (Vol. 3). Cambridge University Press.

Barreto-Souza, W., de Morais, A. L., and Cordeiro, G. M. (2011). The Weibull-geometric distribution. Journal of Statistical Computation and Simulation, Vol. 81(5), pp. 645-657.

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

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

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

Cordeiro, G. M., Alizadeh, M., and 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., and da Cunha, D. C. (2013). The exponentiated generalized class of distributions. Journal of Data Science, Vol. 11(1), pp. 1-27.

Eling, M. (2012). Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models?. Insurance: Mathematics and Economics, Vol. 51(2), pp. 239-248.

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

Frees,E.,Valdez,E.,(1998).Understanding relationships using copulas. North American Actuarial Journal 2, pp. 1–25.

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

Gupta, R. D., and Kundu, D. (1999). Theory & methods: Generalized exponential distributions. Australian & New Zealand Journal of Statistics, Vol. 41(2), pp. 173-188.

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

Kharazmi, O, Saadatinik, A. Hyperbolic cosine-F families of distributions with an application to exponential distribution. accepted.

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

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

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

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, Vol. 82(8), pp. 1191-1206.

Tahir, M. H., Cordeiro, G. M., Alzaatreh, A., Mansoor, M., and 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., and Hamedani, G. G. (2015). The odd generalized exponential family of distributions with applications. Journal of Statistical Distributions and Applications, Vol. 2(1), pp.1.

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

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

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