## 2 Construction of Subdistribution Competing Risk Model with Zero Traumatic Failure Data
### Construction of Subdistribution Competing Risk Model with Zero Traumatic Failure Data
#### 2.2.1 Collection of Product Degradation Information with Zero Traumatic Failure Data  
In order to save time and cost, fixed-time truncated reliability tests are carried out on the samples to monitor and collect performance degradation information. 2.2 Basic Reliability Modeling with Zero Traumatic Failure Data Based on Unilateral Confidence Limit  
Because the Weibull distribution has a wide range of applicability, it is assumed that the tested sample's lifetime follows a two-parameter Weibull distribution with a known shape parameter. According to the relationship between reliability life, total test time, and confidence level under the zero traumatic failure data situation, a reliability function \(R_{B}(t)\) based on the unilateral confidence lower limit method is established. The situation of exponential distribution is analyzed first, and then the situation of Weibull distribution is deduced accordingly. The exponential distribution function and its reliability function are as follows:  
\[\left\{ \begin{array}{l}F(t,\theta)=1-exp\big{[}-\frac{t}{\theta}\big{]}\\ R(t,\theta)=exp\big{[}-\frac{t}{\theta}\big{]}\end{array} \right. \tag{7}\]  
where \(t\) is survival time, \(t>0\); \(\theta\) is an unknown parameter, \(\theta>0\). If a fixed time-truncated test with the fixed time of \(t_{0}\) is carried out on \(n\) products of this type of distribution, the total test time is \(nt_{0}\). According to reference [38], the optimal lower confidence limit of \(\theta\) at the confidence level \(\gamma\) is  
\[\theta_{L}=-nt_{0}/\ln\gamma \tag{8}\]  
Figure 2: The process of the construction of subdistribution competing risk model with zero traumatic failure data. The optimal lower confidence limit of the reliability \(R(t,\theta)\) at the confidence level \(\gamma\) is  
\[R_{B}(t)=exp\bigg{(}\frac{(\ln\gamma)t}{nt_{0}}\bigg{)} \tag{9}\]  
To the reliability life \(t_{R}(\theta)=\theta\ln\frac{1}{R}\), the optimal lower confidence limit of \(t_{R}(\theta)\) at the confidence level \(\gamma\) is  
\[t_{R_{L}}=\frac{nt_{0}(\ln R)}{\ln\gamma} \tag{10}\]  
If the product life obeys a Weibull distribution with shape parameter \(m(>0)\) and scale parameter \(\eta(>0)\), which is written as \(T\sim Weibull(m,\eta)\), the product reliability function is  
\[R(T)=exp\bigg{(}-\bigg{(}\frac{T}{\eta}\bigg{)}^{m}\bigg{)}(T>0) \tag{11}\]  
Let \(t=T^{m},\theta=\eta^{m}\), then, the product life \(t\) obeys an exponential distribution with the parameter \(\theta=\eta^{m}\). If a fixed time-truncated test with the fixed time \(t_{0}\) is carried out on \(n\) products of this type of Weibull distribution, the total test time is \(nt_{0}\). The optimal lower confidence limit of \(\theta\) at the confidence level \(\gamma\) is as follows:  
\[\theta_{L}=\eta_{L}{}^{m}=-\frac{nt_{0}^{m}}{ln\gamma} \tag{12}\]  
If the shape parameter \(m(m>0)\) of the Weibull distribution is known, the optimal lower confidence limit of the reliability \(R\) at the confidence level \(\gamma\) is  
\[R_{B}(t)=exp\bigg{(}\frac{ln(\gamma)}{n}\bigg{(}\frac{t}{t_{0}}\bigg{)}^{m} \bigg{)} \tag{13}\]  
2.3 Regression Coefficient Calculation of Subdistribution Competing Risk Model Based on Performance Degradation Data  
The lower confidence limit of product reliability under the two-parameter Weibull distribution at the confidence level \(\gamma\) with zero traumatic failure data obtained in Equation (13) is used as the basic reliability function \(R_{0}(t)\). Substituting covariate optimal degenerate model \(X(t)\) to the covariate vector.