## 1 Introduction  
PCCS is one of the critical components of the railway pantograph. It collects the electric current from the catenary and supplies to the train. The catenary slides on the surface of the PCCS from one side to another side when the train runs on the track, which generates the wear on the PCCS, and it should be replaced by a new one when the thickness of the PCCS reaches a lower limit. The PCCS needs to be inspected and polished periodically. Since the wear rate for each PCCS is quite different (as illustrated in Section 2), the inspection of the wear condition and the measurement of thickness for the PCCS are time-consuming. Due to the rapid development of the subway, especially in China, the cost control and inventory management of the PCCS have also received considerable attention from the subway operation companies in recent years. Therefore, the prediction of RUL is critical significance to facilitate predictive maintenance. Moreover, the control for the inventory of the PCCS can be further improved through a precise prediction of RUL of the PCCS. The RUL prediction methods have been studied extensively in the last decades. In [1], RUL prediction methods are classified into three categories: physical model method, data-driven methodand the combined method. For the data-driven approach, the typical methods include artificial intelligence methods such as artificial neural network, support vector machine (SVM) and Bayesian methods [2,3]. These methods can achieve sound performance for the RUL prediction in some cases. However, they cannot obtain the probability density function (PDF) of the predicted RUL. Another class of data-driven methods applies statistics to predict the RUL, which has attracted considerable interest in the last decade. [PERSON] et al. [4] summarized the main research results of RUL prediction in recent years, and point out that the current study on RUL prediction should mainly focus on the prediction of the PDF and its expectation. The stochastic process-based method is under the framework of probability theory, and it is one of the conventional methods to estimate the RUL of equipment. This class of methods can obtain the PDF of the RUL prediction and the prediction uncertainty can be analyzed at the same time [5]. The Wiener-process-based method is stochastic process-based method and is also widely-used RUL prediction method. The RUL prediction of this class of approach is updated online via real-time monitoring information of the device. [PERSON] et al. [6] used the linear transformation of the exponential degradation process to establish a linear degradation model. The Bayesian method was used to update the model parameters according to real-time observed bearing degradation data andthus the approximate solution of the RUL distribution of the bearing was achieved. In [7], a model for RUL estimation based on Brownian motion (BM) adaptive drift was developed to compare with other models, and the result shows that the prediction accuracy of the proposed model is improved. The method was further applied to a continuous stirred tank reactor. [PERSON] et al. [8] proposed a BM-driven linear degradation model to implement the adaptive prediction of the RUL, where the Kalman filter (KF) was used to update the drift coefficient, and the expectation maximization (EM) algorithm was proposed to update model parameters. The explicit expression of the RUL distribution was deduced in this paper. [PERSON] et al. [9] considered the RUL prediction of LEDs in the railway carriage and reduced the uncertainty of the prediction. An accelerated degradation model was developed based on the Wiener process, where the parameters are updated by Bayesian inference. [PERSON] et al. [10] proposed a RUL prediction model based on the Wiener process and inverse Gaussian distribution. A real-time RUL estimation of wind turbine bearings was undertaken via the proposed model. [PERSON] et al. [11] incorporated the inspection impact into the degradation modeling based on the Wiener process.