The Beta-Gompertz Distribution

TL;DR

This paper introduces the Beta-Gompertz distribution, a flexible four-parameter model for survival analysis that encompasses several known distributions, with mathematical properties, estimation methods, and real data application.

Contribution

The paper proposes a new four-parameter Beta-Gompertz distribution, extending existing models with enhanced flexibility and analytical properties for survival and reliability analysis.

Findings

The distribution can model various failure rate shapes including decreasing, increasing, and bathtub.

Mathematical expressions for density, distribution, and moments are derived.

Simulation studies confirm the effectiveness of parameter estimation methods.

Abstract

In this paper, we introduce a new four-parameter generalized version of the Gompertz model which is called Beta-Gompertz (BG) distribution. It includes some well-known lifetime distributions such as beta-exponential and generalized Gompertz distributions as special sub-models. This new distribution is quite flexible and can be used effectively in modeling survival data and reliability problems. It can have a decreasing, increasing, and bathtub-shaped failure rate function depending on its parameters. Some mathematical properties of the new distribution, such as closed-form expressions for the density, cumulative distribution, hazard rate function, the $k$th order moment, moment generating function, Shannon entropy, and the quantile measure are provided. We discuss maximum likelihood estimation of the BG parameters from one observed sample and derive the observed Fisher's information matrix. A simulation study is performed in order to investigate this proposed estimator for parameters. At the end, in order to show the BG distribution flexibility, an application using a real data set is presented.