The Marshall–Olkin generalized defective Gompertz distribution for surviving fraction modeling

Inherently, the survival function of a distribution converges to zero as time approaches infinity. The medical interpretation of this definition is that all the patient populations are susceptible to the event of interest that is death or recurrence of the disease after being treated, which is not always the case. This is what prompted the idea of inputting some modification to the domain of the unknown parameters. This way, the survival function is no longer zero when t tends to infinity, it converges instead to the cure rate fraction θ. In this article, we introduce a three-level generalization of the Gompertz distribution for cure rate modeling. This distribution is called the Marshall–Olkin generalized defective Gompertz distribution (MO-GDGD). The main advantage of this new distribution is that it has an increasing, decreasing, constant, and bathtub-shaped failure rate curve. Another strength of this distribution is that it takes into consideration both cases of presence and absence of a cure fraction. Special cases of the model are generated. Maximum likelihood estimates are derived. The applicability of the model is proved using simulated and real data. Statistical tests are used to prove the superiority of MO-GDGD against other models.