A Better Alternative to the Generalized Bilal Distribution: A New Model and Applications

In 2017, A. M. Abd-Elrahman [A new two-parameter lifetime distribution with decreasing, increasing or upside-down bathtub-shaped failure rate, Commun. Stat. - Theory Methods 46 (2017) 8865–8880, doi:10.1080/03610926.2016.1193198] introduced a generalization of the Bilal distribution, where a new two-parameter distribution, generalized Bilal distribution (GBD), was presented. He showed that its failure rate function can be upside-down bathtub-shaped. The failure rate can either be decreasing or increasing due to some mathematical and statistical reasons, which will be given below. In this paper, we introduce a simple and better alternative to the GBD, which will be denoted by WMD. We show that the WMD is a two-parameter distribution which can fit five different types of data sets with respect to their empirical hazard rate functions. Most properties of the WMD are investigated. Point and interval estimation procedures for the two unknown parameters are presented. The existence and uniqueness of the maximum likelihood estimates are proved. The moment estimates are obtained and we showed that one of these estimates is the minimum variance unbiased estimate (MVUE) for its corresponding parameter. A simulation study is provided and the paper is motivated by applications to four different real data sets. A detailed analysis for the Meeker and Escobar data is provided by the book of Meeker and Escobar [Statistical Methods for Reliability Data, 2nd edn. (John Wiley, 1998)]. The results may show that the new distribution provides a better fit than some other most recent existing and already known distributions in the literature. Finally, some concluding remarks are presented.