The The Beta Reduced Modified Weibull Distribution with Applications to Reliability Data

[1] C.D. Lai, M. Xie and D.N.P. Murthy. A modified Weibull distribution. EEE Transactions on Reliability, 52(1):33-37, 2003. [2] M. Bebbington, C.D. Lai, and R. Zitikis. A flexible Weibull extension. Reliability Engineering and System Safety, 92:719-726, 2007. [3] A.M. Sarhan and J. Apaloo. Exponentiated modified weibull extension distribution. Reliability Engineering and System Safety, 112:137-144, 2013. [4] F. Famoye, C. Lee and O. Olumolade. The beta-Weibull distribution. Journal of Statistical Theory and Applications, 4(2):121-136, 2005. [5] L. Benkhelifa. The Weibull Birnbaum-Saunders distribution and its applications. Statistics, Optimization and Information Computing, 9(1):61-81, 2021. [6] W. Nelson. Accelerated life testing: statistical models. data analysis and test plans. New York: Wiley; 1990. [7] G.O. Silva, E.M. Ortega and G.M. Cordeiro. The beta modified Weibull distribution. Lifetime Data Analysis, 16(3):409-430, 2010. [8] G.M. Cordeiro, E.M. Hashimoto and E.M. Ortega and . The McDonald Weibull model. Statistics, 48(2):256-278, 2014. [9] N. Singla, K. Jain and S. Kumar Sharma. The beta generalized Weibull distribution: properties and applications. Reliability Engineering and System Safety, 102:5-15, 2012. [10] A. Saboor, H.S. Bakouch and M.N. Khan. Beta Sarhan–Zaindin modified Weibull distribution. Applied Mathematical Modelling, 40: 6604, 2016. [11] B. He and W. Cui. An additive modified Weibull distribution. Reliability Engineering and System Safety, 145:28-37, 2016. [12] F. Prataviera E.M. Ortega, G.M. Cordeiro, R.R. Pescim and B.A.W. Verssania. A new generalized odd log-logistic flexible Weibull regression model with applications in repairable systems. Reliability Engineering and System Safety, 176:13–26, 2018. [13] A.A. Ahmad and M.G.M. Ghazal. Exponentiated additive Weibull distribution. Reliability Engineering and System Safety, 193:106663, 2020. [14] S.J. Almalki and J. Yuan. The new modified Weibull distribution. Reliability Engineering and System Safety, 111:164-170, 2013. [15] S.J. Almalki. Reduced new modified Weibull distribution. Communications in Statistics - Theory and Methods, 47:2297-2313, 2018. [16] W. Kuo and M.J. Zuo. Reduced new modified Weibull distribution. Optimal reliability modeling: principles and applications. Wiley; 2001. [17] N. Eugene and F. Famoye. Beta-normal distribution and its applications. Communications in Statistics - Theory and Methods, 31:497-512, 2002. [18] B.C. Arnold and H.N. Nagarajah. A first course in order statistics. New York: John Wiley, 2008. [19] I.S. Gradshteyn and I.M. Ryzhik. Table of integrals, Series and Products. Academic Press, New York; 2000. [20] R.G. Miller, G. Gong and A. Muñoz. Survival analysis. New York: John Wiley and Sons; 1981. [21] M.V. Aarset. How to identify a bathtub hazard rate. EEE Transactions on Reliability, 36(1):106-108, 1987. [22] W.Q. Meeker and L.A. Escobar. Statistical methods for reliability data. New York: Wiley; 1998. [23] Y. Liu and A.I. Abeyratne. Practical applications of bayesian reliability. New York: John Wiley and Sons; 2019.

THE BETA-WEIBULL DISTRIBUTION ON THE LATTICE OF INTEGERS

In this paper, a discrete analog of the beta-Weibull distribution is studied. This new distribution contains several discrete distributions as special sub-models. Some distributional and moment properties of the discrete beta-Weibull distribution as well as its order statistics are discussed. We will show that the hazard rate function of the new model can be increasing, decreasing, bathtub-shaped and upside-down bathtub. Estimation of the parameters is illustrated and the model with a real data set is also examined.

Closed form expressions for moments of the beta Weibull distribution

The beta Weibull distribution was first introduced by Famoye et al. (2005) and studied by these authors and Lee et al. (2007). However, they do not give explicit expressions for the moments. In this article, we derive explicit closed form expressions for the moments of this distribution, which generalize results available in the literature for some sub-models. We also obtain expansions for the cumulative distribution function and Rényi entropy. Further, we discuss maximum likelihood estimation and provide formulae for the elements of the expected information matrix. We also demonstrate the usefulness of this distribution on a real data set.