The fractional discrete Weibull distribution

The two-parameter discrete Weibull distribution is an important model especially in reliability studies when the data are reported on a discrete scale‎. ‎The hazard rate function of a discrete Weibull distribution is monotonically increasing and decreasing‎. ‎The present paper provides a family of parametric discrete distributions which is an infinite mixture of exponentiated discrete Weibull distributions‎, ‎and versatile in fitting increasing‎, ‎decreasing‎, ‎and bathtub-shaped failure rate models to different discrete life-test data‎. ‎Some important distributional properties of the model such as the moments‎, ‎order statistics‎, ‎and infinite divisibility are investigated and the parameters of the distribution are estimated by the maximum likelihood method‎. ‎In addition‎, ‎a real data set is analyzed to show the effectiveness of the model‎. ‎Finally we conclude the paper.