Abstract

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.

摘要

本质上，随着时间接近无穷大，分布的生存函数收敛到零。该定义的医学解释是，所有患者群体都容易受到感兴趣的事件的影响，即在接受治疗后死亡或疾病复发，但情况并非总是如此。这就是促使对未知参数的域进行一些修改的想法的原因。这样，当t趋于无穷大时，生存函数不再为零，而是收敛到治愈率分数θ. 在本文中，我们介绍了用于固化率建模的 Gompertz 分布的三级推广。这种分布称为 Marshall-Olkin 广义缺陷 Gompertz 分布 (MO-GDGD)。这种新分布的主要优点是它具有递增、递减、恒定和浴缸形的故障率曲线。这种分布的另一个优点是它考虑了存在和不存在固化部分的情况。生成模型的特殊情况。导出最大似然估计。使用模拟数据和真实数据证明了模型的适用性。统计测试用于证明 MO-GDGD 相对于其他模型的优越性。