Some new results for the Kumaraswamy modified Weibull distribution

Abstract

We study some mathematical properties of the Kumaraswamy modified Weibull distribution pioneered by Cordeiro et al. [4] not discussed by these authors. This model is quite flexible for analyzing positive data since it contains as special models some widely-known distributions, such as the KumaraswamyWeibull, generalized modified Weibull, exponentiated Weibull, modified Weibull and Weibull distributions, among several others. The beauty and importance of this distribution lies in its ability to model both monotone and non-monotone failure rates that are quite common in lifetime problems and reliability. We derive a useful power series for the quantile function. Various new explicit expressions are derived for the asymptotes and shapes, skewness and kurtosis based on the quantile function, the ordinary, incomplete and factorial moments, generating func- tion, and Bonferroni and Lorenz curves. We verify the performance of the maximum likelihood estimates of the model parameters by Monte Carlo simulation. The current model is modified to cope with possible long- term survivors in the data. An application is presented to show the potentiality of this model. A multivariate generalization is proposed.