## 2 Construction of Subdistribution Competing Risk Model with Zero Traumatic Failure Data
### Reliability Model Obtained by the Modified Maximum Likelihood Estimation Method  
Reference [30] assumes that the spindle's life follows the Weibull distribution. According to the degradation data and the modified maximum likelihood estimation method, reference [30] obtained the shape parameter \(m=1.6472\) and the scale parameter \(\eta=2494\); therefore, the reliability function is  
\[R(t)=exp\Big{(}-2.5388\times 10^{-6}t^{1.6472}\Big{)} \tag{18}\]  
Construction of Subdistribution Competing Risk Model for Motorized Spindle with Zero Traumatic Failure Data  
Because of the invariance of the product failure mechanism, in this paper's model, the basic reliability functions of the motorized spindle also obey the Weibull distribution with a shape parameter of 1.6472, let the confidence level \(\gamma=0.90\), \(n=1\). Substituting \(\gamma\) and \(n\) into Equation (13) to calculate basic reliability functions:  
\[R_{B}(t)=exp\bigg{(}-\frac{t^{1.6472}}{1988463}\bigg{)}\]  
Since \(R_{B}(t)\) is regarded as basic reliability \(R_{0}(t)\), substituting \(R_{B}(t)\) into Equation (14), we can obtain the subdistribution competing risk reliability model as follows:  
\[R(t|X(t))=exp\bigg{(}-\frac{t^{1.6472}}{1988463}\bigg{)}^{exp(\beta X(t))} \tag{19}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Time/h** & **Degradation Test Data** & **Time/h** & **Degradation Test Data** \\ \hline  
(\dots\) & \(\dots\) \\ \hline \end{tabular}
\end{table}
Table 1: Performance degradation test data of motorized spindle (axial end runout) \(\upmu\)m.  
Let axial end runout degradation data be the covariate. Put covariates degradation data, degradation time, and degradation status in SPSS; the regression coefficient of the covariate can be calculated by SPSS software(version 22). The covariant coefficients of the spindle are shown in Table 2.  
In Table 2: \(\beta\) is the regression coefficient of the covariate, the regression coefficient of the spindle's axial end runout is \(\beta_{X}=0.185\); \(df\) is the number of the covariates; in this paper, we take only one covariate into account, \(df=1\); \(EXP(\beta)\) reflects the change of failure rate for each unit variable added to the covariate. It means when the axial end runout is added 1 \(\upmu\)m, the failure rate will change 1.203 times. 90% \(EXP(\beta)\) of CI indicates the confidence interval of \(EXP\) (\(\beta\)) with a confidence level of 90%.  
From Table 2, we know that the \(-2\)log likelihood values are smaller than the \(-2\)log likelihood value of the no-covariates model, respectively, and \(p\) values are smaller than 0.05, respectively. This means that the covariate of the risk model has a significant effect on the dependence variable.  
Substituting \(\beta_{X}\) into Equation (19), the reliability models based on the subdistribution competing risk model with zero traumatic failure data can be obtained as follows:  
\[R(t|X(t))=exp\bigg{(}-\frac{t^{1.6472}}{1988463}\bigg{)}^{exp(0.185X(t))}=exp \bigg{(}-\frac{t^{1.6472}}{1988463}\bigg{)}^{exp(0.185X\times 3.876t^{0.088 8})} \tag{20}\]  
The reliability curves of \(R(t|X(t))\) and \(R(t)\) are shown in Figure 3, and the difference curve between the two reliability is shown in Figure 4.  
\begin{table}
\begin{tabular}{c c} \hline
**Axial End Runout** & **Calculation Results** \\ \hline \(\beta\) & 0.185 \\ \(df\) & 1 \\ \(EXP(\beta)\) & 1.203 \\ \(p\) & 0.01 \\ \(-2\)log & 31.503 \\ \(-2\)log of no-covariates & 66.435 \\ \(90\%\)\(EXP(\beta)\) of CI lower & 1.096 \\ \(90\%\)\(EXP(\beta)\) of CI upper & 1.306 \\ \hline \end{tabular}
\end{table}
Table 2: Covariate regression coefficients.  
Figure 3: Reliability curve of \(R(t|X(t))\) and \(R(t)\).