We present competing risks models within a semi-Markov process framework via the semi-Markov phase-type distribution. We consider semi-Markov processes in continuous and discrete time with a finite number of transient states and a finite number of absorbing states. Each absorbing state represents a failure mode (in system reliability) or a cause of death of an individual (in survival analysis). This is an extension of the continuous-time Markov competing risks model presented in Lindqvist and Kjølen (2018). We derive the joint distribution of the lifetime and the failure cause via the transition function of semi-Markov processes in continuous and discrete-time. Some examples are given for illustration.

我们通过半马尔可夫阶段类型分布在半马尔可夫过程框架内展示竞争风险模型。我们考虑在连续和离散时间内具有有限数量的瞬态和有限数量的吸收态的半马尔可夫过程。每个吸收状态代表一个故障模式（在系统可靠性方面）或一个人的死亡原因（在生存分析中）。这是Lindqvist和Kjølen（2018）中提出的连续时间马尔可夫竞争风险模型的扩展。我们通过连续和离散时间中的半马尔可夫过程的跃迁函数得出寿命和失效原因的联合分布。给出一些例子以说明。