Hazard functions play a central role in survival analysis, offering insight into the underlying risk dynamics of time to event data, with broad applications in medicine, epidemiology, and related fields. First order ordinary differential equation (ODE) formulations of the hazard function have been explored as extensions beyond classical parametric models. However, such approaches typically produce monotonic hazard patterns, limiting their ability to represent oscillatory behavior, nonlinear damping, or coupled growth decay dynamics. We propose a new statistical framework for modeling and simulating hazard functions governed by higher-order ODEs, allowing risk to depend on both its current level, its rate of change, and time. This class of models captures complex time dependent risk behaviors relevant to survival analysis and reliability studies. We develop a simulation procedure by reformulating the higher order ODE as a system of nonlinear first order equations solved numerically, with failure times generated via cumulative hazard inversion. Likelihood based inference under right censoring is also developed, and moment generating function analysis is used to characterize tail behavior. The proposed framework is evaluated through simulation studies and illustrated using real world survival data, where oscillatory hazard dynamics capture temporal risk patterns beyond standard monotone models.

翻译：风险函数在生存分析中扮演核心角色，能够揭示事件时间数据背后的风险动态机制，在医学、流行病学及相关领域具有广泛应用。一阶常微分方程（ODE）形式的风险函数模型已被探索作为经典参数模型的扩展。然而，此类方法通常仅能生成单调的风险模式，难以表征振荡行为、非线性阻尼或耦合增长衰减动态。本文提出一种新的统计框架，用于建模和仿真由高阶ODE控制的风险函数，使风险值可同时依赖于当前水平、变化率及时间变量。该模型类别能够捕捉生存分析与可靠性研究中相关的复杂时变风险行为。我们通过将高阶ODE重构为非线性一阶方程组进行数值求解，并借助累积风险函数反演生成失效时间，从而建立仿真流程。同时发展了右删失情形下的似然推断方法，并利用矩母函数分析刻画尾部行为。通过仿真研究评估了所提框架的性能，并利用实际生存数据展示其应用价值——其中振荡型风险动态能够捕捉超出标准单调模型的时序风险模式。