# System Reliability Assessment with Imprecise Probabilities
## 1 Introduction  
In recent years, the rapid development of science and technology has made today's engineering systems (e.g., the control system of the unmanned helicopter) more powerful and complex. However, the resulting reliability problems have become increasingly prominent and the losses caused by failures are huge or even catastrophic. Therefore, it becomes even more important to perform system reliability analysis in order to mitigate the risks and to ensure system safety [1; 2; 3; 4].  
Traditionally, reliability analysis is based on large numbers of samples, which are used to perform statistical inferences. However, in many high-tech fields, such as aerospace, only a limited number of failure data samples are available due to the increased component or system reliability, cost constraints, or other reasons. Limited samples lead to the presence of epistemic uncertainty over the distributional parameters, which make the traditional research method based on a large number of samples no longer applicable [5; 6; 7]. Hence, it is of great significance to develop an effective and efficient reliability analysis method for complex systems under limited samples [8; 9].  
In general, the reliability analysis methods for limited samples can be roughly divided into two categories. One is the method based on virtual sample generation, which transform limited sample problems into large sample problems for processing, such as bootstrap [10; 11] and support vector machine methods [12; 13]. The second comprises methods based on Bayesian theory, which combine multiple forms of information for statistical inference. [PERSON] et al. [14] proposed a Bayesian Monte Carlo method for high-reliability relay protectors obeying the Weibull distribution, for which limitedfailure data were available. [PERSON] et al. [15] investigated the estimation procedure for the parameter, reliability, and hazard functions of the inverted exponential distribution under progressive type-II censored data in the Bayesian framework. [PERSON] et al. [16] proposed a generalized inverse Gaussian prior to lognormal linear regression models and discussed model parameter selection under small and medium sample sizes. For the problem that it is difficult to calculate the posterior distribution of complex distribution models, [PERSON] et al. [17] investigated the problem of point and interval estimations for the modified Weibull distribution using a type-II progressively censored sample based on the Markov Chain Monte Carlo (MCMC) approach. [PERSON] et al. [18] derived Bayesian estimations of the hybrid censored lognormal distribution. In addition, [PERSON] et al. [19] wrote a book on the use of Bayesian theory in reliability assessment.  
As is clear from the literature, the major focus of recent research has been on the processing of limited samples to get a reasonable result during reliability analysis of a complex system. In this paper, an efficient and effective method for reliability analysis of complex systems is proposed based on the theories p-box and survival signature. The rest of the paper is organized as follows: Section 2 gives a brief description of the theory of Bayesian interval estimation and p-box. As examples, the p-boxes of the exponential and Weibull distributions are obtained analytically, based on the Bayesian theory. In Section 3, we present a method for the reliability analysis of complex systems with limited samples, based on the theories of p-box and survival signature, while Section 4 gives two numerical examples to illustrate the application of the proposed approach. Finally, Section 5 presents the conclusions of the paper and some ideas for related future work.