The Weibull Negative Binomial Distribution

We propose the Weibull negative binomial distribution that is a quite flexible model to analyze positive data, and includes as special submodels the Weibull, Weibull Poisson and Weibull geometric distributions. Some of its structural properties follow from the fact that its density function can be expressed as a mixture of Weibull densities. We provide explicit expressions for moments, generating function, mean deviations, Bonferroni and Lorenz curves, quantile function, reliability and entropy. The density of the Weibull negative binomial order statistics can be expressed in terms of an infinite linear combination of Weibull densities. We obtain two alternative expressions for the moments of order statistics. The method of maximum likelihood is investigated for estimating the model parameters and the observed information matrix is calculated. We propose a new regression model based on the logarithm of the new RODRIGUES, CORDEIRO, DEMÉTRIO and ORTEGA 26 distribution. The usefulness of the new models is illustrated in three applications to real data.