Weibull-Fréchet distribution: A new lifetime distribution with application to gastric cancer data

Department of Mathematics, Faculty of Mathematics and Natural Sciences (FMIPA), Universitas Indonesia, Depok 16424, Indonesia

^* Corresponding author: snurrohmah@sci.ui.ac.id

Abstract

Lifetime data is a type of data that consists of waiting time until an event occurs. Some of the events of lifetime data are deaths, occurrence of a disease, or failure of a machine. The distribution usually used for modeling lifetime data is the Weibull distribution. However, Weibull distribution has a limitation in its application: it can only model data with a monotonic hazard function. Therefore, a method for generalizing the Weibull distribution is needed so it can model data with a non-monotonic hazard function. One of those generalizations is the Weibull-Fréchet distribution (WFr) which was introduced by Afify in 2016. The WFr distribution has an advantage over the Weibull distribution, due to its capability in modeling data with unimodal hazard function. The method used in generating the WFr distribution is the Weibull-G (WG) that were introduced by Bourguignon in 2014. The WG method combines the distribution of a Weibull distribution with an arbitrary distribution with a cumulative distribution function (cdf) G(x) using a function W[G(x)]. The characteristics of WFr distribution discussed include probability density function (pdf), cumulative distribution function, survival function, hazard function, and the moment. The hazard function of WFr can be monotonic or unimodal. The maximum likelihood estimation method is used in estimating the parameters of the distribution. Finally, lifetime data of gastric cancer patients is given for illustration purposes. The data is modeled using the WFr distribution, and both the Weibull and Fréchet distribution for comparison. The model result shows that the WFr distribution is the best distribution for modeling the lifetime data of gastric cancer patients.