## 2 Construction of Subdistribution Competing Risk Model with Zero Traumatic Failure Data  
### Subdistribution Competing Risk Model  
Competing risk means that when the interested events occurred to the subject we studied, there are other outcome events that may happen at the same time, which will prevent or reduce the occurrence probability of the interested events, forming a so-called \"competing\" relationship among the outcome events; this series of events is called competing events, as shown in Figure 1.  
In Figure 1, assuming that an individual has an initial state and a set of ending events caused by reason \(n(n=1,\ldots,k)\), for the individual, the \(k\) ending events are competing with each other. If the interested event is the ending event caused by reason 1, then other ending events are the competing events of the ending event caused by reason 1. In this case, if the competing events are treated as censored data, there will be a large deviation when estimating the interested event's incidence risk using the Kaplan-Meier method [31]. So, the competing risk model should be used in this situation. The proportional risk model is a kind of competing risk model put forward by [PERSON] [32]. It has been used in the field of medical survival analysis at first. Because it can deal with the relationship between the multivariate and the interest events without knowing the basic failure rate function, recently it has also been applied in many aspects, such as product reliability [33], product failure impact analysis [34], and product maintenance strategy [35]. It is a multivariate regression model, and it takes the influence degree of multi-factors related to product life into account. The model can be constructed by estimating the regression coefficients [36]. [PERSON] proportional risk model treats competing events as censored data, making it overestimate the incidence risk of the interested events, resulting in competing risk bias. Special studies have found that about 46% of the literature may have such bias. In 1999, [PERSON] and [PERSON] proposed a semi-parametric proportional risk model of partial distribution, namely the subdistribution competing risk model, also known as the Fine-Gray Model [37]. In the subdistribution competing risk model, the subdistribution risk of the interested event \(k\) is defined as the probability of failure caused by \(k\) at an infinitesimal time interval \(\Delta t\), on the condition that no event occurred to the individual before time \(t\) or no event different from \(k\) occurred to the individual before time \(t\). The subdistribution risk \(\lambda(t)\) can be expressed as follows:  
\[\lambda(t)=\lim_{\Delta t\to 0}\frac{P(t\leq T<t+\Delta t,D=k|T\geq t \cup\{T<t,D\
eq k\})}{\Delta t} \tag{1}\]  
where \(T\) is the time when competing event \(k\) occurs; \(D=k\) means the failure occurs and is caused by event \(k\). Figure 1: Competing risk model: an initial state and k mutually independent ending events. Cox-based subdistribution competing risk model is  
\[\lambda(t\left|\mathbf{X}\right\rangle=\lambda_{0}(t)\cdot exp\left(\beta^{T} \cdot\mathbf{X}\right) \tag{2}\]  
where \(\lambda(\,t|\mathbf{X})\) is the failure rate function of the interested event at time \(t\); \(\lambda_{0}(t)\) is the basic failure rate function of the interested event at time \(t\); \(\mathbf{X}\) is covariate vector, also called factors that affect the failure rate; \(\beta\) is parameter vector, also called regression coefficient vector. The relationship between failure rate function \(\lambda(\,t|\mathbf{X})\) and the subdistribution distribution function \(F(t|\mathbf{X})\) is shown as follows:  
\[F(t\left|\mathbf{X}\right\rangle=1-exp\left(-\int\limits_{0}^{t}\lambda(t \left|\mathbf{X}\right\rangle)dt\right) \tag{3}\]  
Therefore, \(F(t|\mathbf{X})\) can be directly obtained from the estimation value of the regression covariate coefficient \(\beta\) and the basic failure rate function \(\lambda_{0}(t)\).