## 1 Introduction  
Space TWTA is a key component of spacecraft transponders and spacecraft transmitters. It is mainly used in the payload of military communication satellites, navigation and positioning satellites, microwave remote sensing satellites, and so on, to realize the function of microwave power amplification [1]. As the core component of the satellite key subsystem, space TWTA is particularly important in aerospace applications, and its life directly affects the satellite's service life and in-orbit stable operation. Therefore, it is of great significance to study the in-orbit life of TWTA for the Prognostics and Health Management (PHM) of satellites. In general, commonly used reliability prediction methods are divided into two main categories: failure mechanism-driven methods and data-driven methods. The failure mechanism-driven model is used to establish a degradation model through the relationship between the failure modes selected by the internal failure mechanism of the product and a large amount of historical operation data. However, the data-driven methods are used to predict the probability density function (PDF) and cumulative distribution function (CDF) of the product's life through its historical failure data and the existing observation data. These methods do not rely on the operating principle of the product and the inherent failure mechanism, mainly including neural networks, support vector machines, state-space models, and stochastic processes. However, due to the high manufacturing cost and complex failure mechanisms of space TWTA, a single-cathode degradation model [2] cannot accurately reflect the degradation of TWTA, and it is also difficult to establish the physical failure model of the entire tube. Therefore, there are several difficulties in applying the prediction methods based on the failure mechanism models in the engineering application of the reliability evaluation of the TWTA. Performance degradation-based methods are one of the essential data-driven methods based on the fact that the hidden life information is reflected by one or multiple performance characteristics, and they have been recognized as an effective and important approach for the reliability evaluation of the high-reliability and long-life products. They also provide a reference to carry out the research on PHM of TWTA. Many research results and theories have been obtained for reliability evaluation based on performance degradation methods, such as degenerate trajectory models, degenerate distribution models, and stochastic processes [3; 4; 5; 6; 7; 8]. In comparison with the first two models, stochastic process-based methods are more in line with engineering practice because they can describe the uncertainties of performance degradation and the individual differences between different units. In engineering applications, most scholars mainly focus on one-dimensional stochastic processes. However, there are multiple degradation trajectories or components for a complex system. It may make the product or equipment reliability analysis fail to obtain relatively accurate results or even lead to rigorous conclusions [9], taking no consideration of their correlation with each other. Many experts and scholars have introduced the Copula function [10] to deal with relevant issues, for instance, the heavy machine tool [11], the lithium-ion battery [12], the rubidium atomic clock [13], and to study the multiple-parameter degradation modeling of complex systems and products because of the characteristics of its simple structure and accurate description of the correlation between failure modes and random variables, and have achieved fruitful research results in recent years. For example, [PERSON] et al. [14] used the Copula function to establish the joint PDF between multi-dimensional random variables for complex and multi-dimensional-related mechanical systems and thus proposed a system reliability analysis method. [PERSON] et al.