## 1 Introduction  
For products like motorized spindles, which have high reliability and long life, it is difficult to obtain traumatic failure information in the short term due to constraints such as time or funding. Therefore, the life assessment of such products has always been a research hotspot and difficulty. Because of the characteristics of these products' long life and high reliability, there are very few traumatic failures in their whole life cycle. Therefore, many references have conducted research on product reliability modeling and life assessment methods with zero traumatic failure information [1; 2; 3; 4]. [PERSON] and [PERSON] [5] presented a method of reliability analysis of the time-truncated zero-failure data based on the Weibull distribution under the zero-failure test data circumstances. [PERSON] et al. [6] solved the problem that traditional maximum likelihood estimation (MLE) cannot be used to estimate the parameters of a reliability model under zero-failure data and proposed a modified MLE method. [PERSON], [PERSON] et al. [7] recommended the Bayes method to deal with zero-failure data. However, the key to Bayes' application is to obtain prior information correctly. [PERSON], [PERSON] et al. [8] introduced the concept of modeling uncovered faults when a product test reveals no failures and applying the statistics of the extremes to estimate the coverage. [PERSON], [PERSON] [9] and [PERSON] et al. [10] proposed that parameters related to product reliability should be used as random variables. They then used zero-failure test data to estimate the parameters of different confidence intervals and obtained the product reliability under a unilateral confidence limit directly. They thus simplified the calculation process of product reliability and improved the efficiency of evaluation. [PERSON] [11] used the unilateral confidence limit method to calculate the confidence limits of relevant parameters under exponential distribution and Weibull distribution; however, the test sample size under the Weibull distribution affects the true value of the shape parameter, which in turn affects the product reliability. Therefore, [PERSON] et al. [12] proposed a modified unilateral confidence limit evaluation method by introducing failure information into the confidence limit analysis. With the introduction of failure information, the Bayesian estimator of product failure rate was obtained, then the product reliability and MTBF were obtained by using the failure rate estimator and the modified unilateral confidence limit method. However, the above-mentioned research only studied the reliability of products with zero-failure data from the perspective of traumatic failure, ignoring the impact of product performance degradation on product reliability, so the research results are biased. As the performance of products always degrades during the period of use, the degradation information is easily available. So, the product reliability study is usually carried out in combination with the degradation information [13; 14; 15]. However, some scholars found that the degradation process of products often comes with traumatic failure. In general, the more degraded the performance, the more likely it is to be accompanied by traumatic failure. The failure phenomenon with both traumatic failure and degradation failure is called competing failure [16]. Therefore, [PERSON] et al. [17] established a competing failure reliability evaluation model. At present, a lot of research has found that products are prone to traumatic failure as the performance gradually degrades. Therefore, it better reflects engineering reality to model the reliability and evaluate the product's life from a competing point of view. At present, there are many modeling methods of competing failure [18; 19; 20; 21; 22]. [PERSON] et al.