# Reliability Analysis of Small-Sample Failure Data for Random Truncation High-Voltage Relay
###### Abstract  
In order to model and evaluate the reliability of long-life high-voltage relays with small-sample fault data characteristics, a reliability analysis method integrating average rank, the minimum mean square distance empirical distribution function, and total least squares estimation is proposed. In the random truncation experiment, considering the influence of random truncation data, the average rank method is used to correct the rank of small-sample fault data; then, the optimal empirical distribution function for small-sample fault data is obtained through the minimum average square distance, which can overcome the impact of small-sample fault data randomness. Under the assumption of the Weibull distribution model, the total least squares estimation method is used for reliability model parameter estimations, and the linear correlation coefficient and d-test method are used for model hypothesis testing. If two or more distribution models pass the linear correlation coefficient test and the d-test simultaneously, the root mean square error and relative root mean square error are applied to determine the optimal reliability model. The effectiveness of this method is verified by comparing it with the maximum likelihood estimation method. high-voltage relay; reliability evaluation; small sample; random truncation; average rank method +
Footnote †: journal:  
0000-0001-9999-4000-0000-0000-0000-0000-0000-0000-0000-0000-0  
usually uses prior information of similar products and combines prior information with small-sample data for sample expansion. However, for new products with fast updates, it is difficult to have sufficient and accurate empirical data to utilize. The Bayesian estimation method uses historical information or prior information of similar products to estimate the parameters of the reliability model and corrects the parameters to obtain the posterior distribution of the reliability model parameters through the obtained small-sample data, thereby conducting a reliability evaluation [6,7]. [PERSON] [8] pointed out that the selection of Bayesian prior distribution in small-sample situations has a very sensitive impact on the evaluation results. Machine learning methods such as neural networks [9] and support vector machines (SVMs) [10] are also used to analyze reliability under a small-sample-data situation. However, neural networks have problems such as a slow convergence speed, inconsistent network structure selection, and local optimal solutions. SVMs are extremely sensitive to sample errors and lack adaptive ability, which can greatly reduce the robustness of reliability analyses. The parameter estimation methods for reliability distribution models can be divided into the moment method, the maximum likelihood method, the differential evolution method, the Bayesian method, the grey model method, and the least squares estimation method [11,12]. Most of these methods do not have analytical solutions and require solving nonlinear equations or the maximum points of multivariate functions. The grey model method can estimate the three parameters of three-parameter Weibull distribution at once without iteration, and it avoids the problem of correlations existing between parameters and has a fast calculation speed, making it suitable for small-sample Weibull distribution data analysis [13]. However, the principle of this method limits the order of its fault data to integers, so it is not suitable for small-sample situations with missing data. The least squares estimation method is widely used due to its simple principle and analytical solution. It also has the advantage of being independent of the initial values. The assumption distribution model in engineering practice is usually a two-parameter Weibull distribution [14].