## 1 Introduction
Also, considering the fault prediction and fault diagnosis, a health management system was established to evaluate the system reliability of power servo equipment [19]. According to geometric and vibration errors, a machining accuracy reliability evaluation method was investigated, based on the milling dynamics theory and the full-discretization method [20]. Considering the machining error out of tolerance, an evaluation method of accuracy reliability was proposed under different machining loads [21]. Accuracy reliability [22] and transmission error [23] are regarded as two of the most important indices to evaluate system performance. Both are influenced by the transposition system of the power servo tool holder. Qualitative and quantitative methods are utilized to investigate the reliability of the transposition system in power servo tool holders. Considering various factors, different indices and methods are analyzed for system accuracy reliability based on the use of quantitative methods. The moment-based method for reliability evaluation attracts researchers' attention. This method relies on the moment information of input variables rather than the probability distribution functions. The moment-based method can solve multi-dimensional reliability problems and improve the accuracy with low information [24]. The first two order moments of calculation error [25] can be estimated according to the statistical moment similarity for reliability analysis. The probability density function (PDF) of a performance function can be constructed from the perspective of the first four statistical moments [26; 27]. Also, the fitting method is utilized for the descriptions of high moments in order to evaluate system reliability [28]. The properties of statistical moments of standard normal distribution parameters are utilized to improve the high-order statistical moment method for the stable and accurate calculation of structural reliability analysis with insufficient data [29]. The first four moments of random input variables are derived to improve the accuracy of the reliability prediction based on the statistical properties of the standard normal random variable [30]. The first three statistical moments and fault trees [31] are combined to optimize resource allocations for reliability improvement. An improved high-order moment-based saddle-point approximation method for reliability analysis is presented [32],and the reliability-based sensitivity [33] has been investigated to quantify the influences of different parameters in engineering. Also, the maximum entropy and different order moments [34] are utilized for the approximation of the reliability-based sensitivity index. The first four statistical moments can be utilized to evaluate system reliability with high-order characteristics and different performances. Also, based on statistical moments, reliability-based sensitivity can present the parameters' influences and changing laws in engineering, which supports the need for system optimization and equipment upgrading. Reliability evaluation and reliability-based sensitivity are investigated for the transposition system in the power servo tool holders of CNC. Based on statistical moments, the reliability method is proposed for accurately measuring system performance with random effects. Also, the reliability-based sensitivity with high-order moments is proposed to analyze the effects of different parameters in complex system analysis and design. This method is utilized to assess the system accuracy of the transposition system in the power servo tool holder. In Section 2, responses of the transposition system are investigated. The first four statistical moments are derived to depict the system characteristics in Section 3. Reliability and reliability-based sensitivity evaluation methods are proposed based on statistical moments in Sections 4 and 5, respectively. Different methods are compared for their computational accuracy and rationality.