On the Exponentiated Unit Gompertz Distribution and its Bivariate Extension

In this article, we introduce an extension for the Gompertz distribution, which is called the exponentiated unit Gompertz distribution depending on the proportional hazard model. The density, survival, hazard, and reversed hazard functions. In addition, we introduce the moments, quantiles, mode, order statistics, and joint order statistics. We estimate the unknown parameters of the new distribution by maximum likelihood estimation, and an approximate variance-covariance matrix has been obtained. Some simulations have been performed to see the performance of the MLEs. Depending on three data sets, we compared the fitting of the exponentiated unit Gompertz distribution with some lifetime distributions that are used to describe bounded data. By using the Farlie–Gumbel–Morgenstern coupla, we introduce a new bivariate exponentiated unit Gompertz distribution and study its statistical properties. Maximum likelihood and inference functions for margins approaches are used to estimate the bivariate model's unknown parameters. Finally, we illustrate the importance of the proposed bivariate model using a real data set in the medical field.