## 1 Introduction
In this way, the ordered lifetime data for \(m\) elements are obtained using the censoring scheme \(\{M,\ m,\ R_{1},\ R_{2},\...,\ R_{m}\}\). This process is shown in Figure 1. In particular, progressive Type-II censoring can be seen as an extension of Type-II censoring, which is derived by assuming \(R_{i}=0,\ i=1,\ 2,\...,\ m-1\). Otherwise, the complete sample is derived by assuming \(R_{i}=0,\ i=1,\ 2,\...,\ m\). For samples that have been censored or are incomplete, there are few studies on the inference of the MSSR in the literature. [PERSON] et al. [9] studied the estimation of the MSSR for Burr XII distribution using progressive first-failure censoring data. [PERSON] et al. [10] discussed the estimation of the MSSR for generalized exponential distribution using Type-I censoring data. [PERSON] et al. [11] obtained the estimation of the MSSR for Topp-Leone distribution using progressive censoring data. Many probability distributions have been proposed to fit the lifetime data. Here, we select the Chen distribution proposed by [PERSON] [12]. Its probability density function (PDF), CDF, and hazard rate function (HRF) are, respectively, given by  
\[f(x;\ \theta,\ \lambda)=\theta\lambda x^{\lambda-1}\exp\{x^{\lambda}+\theta(1- e^{x^{\lambda}})\}, \tag{2}\]  
\[F(x;\ \theta,\ \lambda)=1-\exp\{\theta(1-e^{x^{\lambda}})\}, \tag{3}\]  
\[h(x;\ \theta,\ \lambda)=\theta\lambda x^{\lambda-1}\exp\{x^{\lambda}\},\]  
where \(x>0\) and the shape parameters \(\theta,\ \lambda>0\). Hereafter, the Chen distribution can be represented by \(\text{Chen}(\theta,\ \lambda)\). Images of the PDFs and HRFs of the Chen distribution for certain cases are presented in Figure 2. It can be seen that the Chen distribution has a flexible PDF and HRF when the shape parameters are taken at different values. When  
Figure 1: A schematic diagram of progressive Type-II censoring. \(\lambda>1\), the PDFs have a significant peak; when \(\lambda=1\), the PDFs have a roughly decreasing trend; and when \(0<\lambda<1\), the PDFs are monotonically decreasing. Additionally, when \(\lambda\geq 1\), the HRFs are monotonically increasing, and when \(0<\lambda<1\), the HRFs present as a bathtub-shaped curve. In the literature, the Chen distribution has received considerable attention and has been studied by several scholars, including [PERSON] and [PERSON] (2013), [PERSON] et al. (2014), [PERSON] et al. (2015), [PERSON] et al. (2016), and the references therein. Let \(X_{1},\ X_{2},\...,\ X_{k}\) follow \(Chen(\theta_{1},\ \lambda)\) independently, and let \(Y\) (independent of \(X_{1},X_{2},...,X_{k}\)) follow \([PERSON](\theta_{2},\ \lambda)\). They have a common second shape parameter and different first shape parameters. Now, based on Equations (1)-(3), \(R_{s,k}\) can be derived as  
\[ \begin{array}{ll}R_{s,k}&=\theta_{2}\lambda\sum_{m=s}^{k}\left(\begin{array} []{c}k\\ m\end{array}\right)\int_{0}^{\infty}x^{\lambda-1}exp\Big{\{}\Big{(}1-\varepsilon ^{x^{\lambda}}\Big{)}(\theta_{1}m+\theta_{2})+x^{\lambda}\Big{\}}\Big{\{}1- exp\Big{\{}\theta_{1}\Big{(}1-\varepsilon^{x^{\lambda}}\Big{)}\Big{\}}\Big{\}}^{k-m}dx\\ &=\theta_{2}\sum_{m=s}^{k}\left(\begin{array}{c}k\\ m\end{array}\right)\sum_{n=0}^{k-m}\left(\begin{array}{c}k-m\\ n\end{array}\right)(-1)^{n}\int_{1}^{\infty}exp\{[\theta_{1}(m+n)+\theta_{2}] (1-t)\}dt\\ &=\sum_{m=s}^{k}\sum_{n=0}^{k-m}\left(\begin{array}{c}k\\ m\end{array}\right)\left(\begin{array}{c}k-m\\ n\end{array}\right)\frac{(-1)^{n}\theta_{2}}{(m+n)\theta_{1}+\theta_{2}}, \end{array}  \tag{4}\]  
where \(t=e^{x^{\lambda}}\). As far as we know, no research has been conducted on the inference of the MSSR for the Chen distribution using progressively censored data. [PERSON] et al. (2017) discussed its classical estimation using Type-II censored data. So, our goal is to expand their research by carrying out its estimation from the Chen distribution using progressively censored data. In addition, the Bayesian estimation is also considered. This article is organized as follows. In Section 2, the maximum likelihood estimate (MLE) is studied.