The statistical model employed and the available sample size have a significant effect on the estimation of parameters in reliability assessment. A good statistical model provides a better fit to the actual lifetime data, and a good parameter estimation method can reduce the estimation error. For some time now, the normal distribution, lognormal distribution, exponential distribution, and Weibull distribution have served as commonly employed statistical models in reliability assessment. However, these models can only describe a particular type of failure rate. Lai et al. [
1] developed a new model, denoted as the modified Weibull distribution, which included three parameters to control the shape of the curve, and employed this distribution to describe failure rates over the entire lifetime. Liu et al. [
2] presented a new Bayesian model based on the Weibull distribution using a concave method and the relationships of failure probabilities in time, and verified the stability of the model via a practical application. Singh et al. [
3] derived Bayesian estimations of hybrid censored lognormal distribution compared with maximum likelihood estimations and further computed Fisher information matrix, equal-tail and highest posterior density, and so on. Soliman et al. [
4] studied the point and interval estimations of the modified Weibull distribution under squared error loss and linear exponential loss using the Markov chain Monte Carlo (MCMC) technique as an advanced algorithm to solve the problem of complex and high-dimensional integration for progressively type-II censored samples, and compared the results obtained based on two real data sets with those obtained using maximum likelihood estimation (MLE). Yang et al. [
5] proposed a Bayesian reliability model under conditions of small sample size would have been applied to an NC machine tool, and a different method from that of MCMC would have been employed for high-dimensional integration. Xia [
6] proposed a gray bootstrap method that can generate simulated data to compute reliability without other prior information for small sample size zero-failure data. Lord et al. [
7] proposed a Poisson-gamma model describing motor vehicle crashes using Bayesian inference, and compared the results of numerical simulation with those obtained from MLE to demonstrate that the Bayesian estimation (BE) performed better than MLE. Fabrizi et al. [
8] proposed a generalized inverse Gaussian prior for log-normal linear regression models and discussed how to choose parameters of the models under small and medium sample sizes. Junttila et al. [
9] applied a Bayesian principal component regression model for remotely sensed data and verified it against other methods under spatial effects, multicollinearity and small size using an efficient Markov chain Monte Carlo sampling scheme. Hao et al. [
10] proposed an improve Bayesian approach for precision system reliability assessment through searching for feasible points, screening feasible points and then simplifying the likelihood function. Ming et al. [
11] presented the Bayesian model based on the mixed beta distribution. Comparing with traditional Bayesian analysis, the model presented considers differences between similar products. Jin [
12] applied a bootstrap method for generating data to obtain the prior distribution and hierarchy of a Bayesian model describing the degradation process of space bearings for the purpose of calculating lifetime. A reliability evaluation method for segmented distribution using Bayesian method and an improved information criterion is proposed by Li et al. [
13]. The method are proven more efficient and accurate than traditional method. Jia et al. [
14] verified Bayesian estimation of multiply Type-I censored Weibull distribution is satisfactory for reliability assessment according to Monte Carlo simulation results. Peng et al. [
15,
16] developed a Bayesian model updating approach to integrate subjective information in adjacent periods and in specific periods of lifetime stages, and applied the approach to assess a newly developed gantry-type machining center. Yang et al. [
17] presented a comprehensive reliability allocation method based on cubic transformed functions, which can be designed to emphasize the failure severity or the failure occurrence depending on requirements. Salvinder et al. [
18] employed a Markov chain model to represent the bending and torsion loads and estimated Weibull shape parameter to provide an accurate, efficient, fast and cost effective reliability analysis. Hamada et al. [
19] and Cai et al. [
20] have written relevant monographs regarding the use of Bayesian methods in reliability assessment that treat numerous topics such as advanced computation algorithms, model selection, degradation processes, and reliability growth analysis, and discuss several cases. Yao et al. [
21] proposed a reliability assessment method based on T-S fault tree method and Bayesian network method, and compared with T-S fault tree method and Bayesian network method to prove feasible method.
Considering the hazardous nature of high-speed train operation, the assessment of vehicle reliability is essential to guarantee its safe operation. It is therefore necessary to assess the reliability of key components in high-speed trains, which represent issues that have been widely and thoroughly studied. Meng et al. [
22] researched a reliability evaluation method using Markov model to describe the trend of the system reliability, and assessed the reliability of high-speed train traction transmission system. However, the system reliability needs research component reliability, which is statistical inference under small sample problem. Wang et al. [
23] employed the least squares method to analyze the reliability of the key components of trains based on the Weibull distribution. The results of the study indicated that the sample size had a significant effect on the estimation precision. Wu et al. [
24] assessed the reliability of relay valves based on the characteristic that the failure rate gradually increases according to an assumed Weibull distribution. Wang et al. [
25] performed a durable test to evaluate the reliability of a brake unit for urban rail vehicles. The methods discussed above for the reliability assessment of high-speed trains can achieve satisfactory results under large sample size. Tian [
26] employed a virtual expansion method to expand data sample size from
n = 2–3 to
n = 12, and then employed the method to analyze the reliability of the center sill and body bolster of a C70 gondola car based on a Bayesian method. The authors assumed that the lifetimes of parts followed normal distributions, and the approach was demonstrated to improve the assessment precision. Zhu et al. [
27] extended the Bayesian method to the Weibull distribution, and verified the robustness of the model by assessing the reliability of bearings in a high-speed train. Subjective selection was employed to define the prior distribution and its hyperparameters, and these estimators were solved using an analytical method. Dong [
28] assessed the reliability of braking systems in urban railway vehicles using a goal-oriented (GO) methodology and Bayesian theory based on the assumption of an exponential distribution. The method was demonstrated to be suitable for analyzing conditions where a component follows the rule that the failure rate is constant. Akama [
29] calculated the crack propagation rate of cracks in a Shinkansen vehicle axle using Bayesian analysis by combining crack propagation rates derived from small specimens with crack propagation rates derived from full-scale models. The results showed that the method can narrow the variance of the fatigue life distribution, and provide more confident failure probability values.
Generally, reliability assessments are based on large sample size to attain statistical inferences (usually
n > 30). As such, reliability experiments consume considerable time and money. Owing to the long lifetimes, high cost, and complex structures of components in high-speed trains, it is necessary to develop reliability assessment theory and methods suitable under small sample size. The present study proposes a new approach that combines a Bayesian method with a subjective prior distribution for zero-failure data under small sample size, and experiments confirm that this approach can be safely applied to the reliability assessment of solenoid valves in the braking systems of high-speed trains. In Section
2, the failure model and the characteristics of zero-failure data for the solenoid valve are discussed. In Section
3, a Bayesian reliability model based on the binomial distribution at censored time is developed in detail. In Section
4, a modified Weibull distribution is introduced for the solenoid valve using least squares estimation based on failure probabilities at censored time. In Section
5, a numerical simulation is performed to compare the results of the proposed method with those of MLE. In Section
6, an actual numerical case is analyzed for the solenoid valve of the braking system in high-speed trains. Section
7 provides concluding remarks.