General results for the beta-modified Weibull distribution

Abstract

We study in detail the so-called beta-modified Weibull distribution, motivated by the wide use of the Weibull distribution in practice, and also for the fact that the generalization provides a continuous crossover towards cases with different shapes. The new distribution is important since it contains as special sub-models some widely-known distributions, such as the generalized modified Weibull, beta Weibull, exponentiated Weibull, beta exponential, modified Weibull and Weibull distributions, among several others. It also provides more flexibility to analyse complex real data. Various mathematical properties of this distribution are derived, including its moments and moment generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are also derived for the chf, mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The estimation of parameters is approached by two methods: moments and maximum likelihood. We compare by simulation the performances of the estimates from these methods. We obtain the expected information matrix. Two applications are presented to illustrate the proposed distribution.

Keywords:

• beta distribution
• exponentiated exponential
• exponentiated Weibull
• Fisher information matrix
• generalized modified Weibull
• maximum likelihood
• modified Weibull
• Weibull distribution

Acknowledgements

The authors would like to thank the editor, the associate editor and the referee for carefully reading the paper and for their comments, which greatly improved the paper.