## 2 Maximum Likelihood Estimation  
In this section, we derive the MLE of \(R_{s,k}\) by adapting the approach of [PERSON] et al. (2017) and extending its application to progressively Type-II censored data. Assume that \(N\) systems with \(K\) strength components are subjected to a lifetime test. We observe the lifetime data for \(n\) systems with \(k\) components using progressive Type-II censoring. The observed \(data\underset{\sim}{x}\) and \(\underset{\sim}{y}\) are described as follows:  
\[\text{Observed strength values}\qquad\text{Observed stress values}\]  
\[\left( \begin{array}{cccc}x_{11}&x_{12}&\cdots&x_{1k}\\ \vdots&\vdots&\ddots&\vdots\\ x_{n1}&x_{n2}&\cdots&x_{nk}\end{array} \right)\qquad\text{and}\qquad\left( \begin{array}{c}y_{1}\\ \vdots\\ y_{n}\end{array} \right)\]  
where each row of values in \(\underset{\sim}{x}\) is the censored sample from \([PERSON](\theta_{1},\ \lambda)\) with the progressive Type-II censoring scheme \(\{K,\ k,\ S_{1},\ S_{2},\...,\ S_{k}\}\), and \(\underset{\sim}{y}\) is the censored sample from \([PERSON](\theta_{2},\ \lambda)\) with the progressive Type-II censoring scheme \(\{N,\ n,\ T_{1},\ T_{2},\...,\ T_{n}\}\). For simplicity, we denote the censoring scheme as \(S=(S_{1},\ S_{2},\...,\ S_{k})\) and \(T=(T_{1},\ T_{2},\...,\ T_{n})\). Then, the likelihood function is  
\[L(\theta_{1},\theta_{2},\lambda)=c_{1}\prod_{i=1}^{n}\Biggl{(}c_{2}\prod_{j=1 }^{k}f_{X}\bigl{(}x_{ij}\bigr{)}\bigl{[}1-F_{X}\bigl{(}x_{ij}\bigr{)}\bigr{]} ^{S_{j}}\Biggr{)}f_{Y}(y_{i})\bigl{[}1-F_{Y}(y_{i})\bigr{]}^{T_{i}}, \tag{5}\]  
where  
\[c_{1}=N(N-T_{1}-1)\cdots(N-T_{1}-\cdots-S_{n-1}-n+1),\]  
\[\frac{\partial l}{\partial\theta_{2}}=\frac{n}{\theta_{2}}+\sum_{i=1}^{n}\,(1-e^{y _{i}{}^{\lambda}})(1+T_{i})=0. \tag{9}\]  
Using Equations (8) and (9), we derive  
\[\vec{\theta_{1}}=\frac{nk}{\sum_{i=1}^{n}\sum_{j=1}^{k}\,(e^{x_{ij}{}^{\lambda }}-1)(1+S_{j})},\,\,\vec{\theta_{2}}=\frac{n}{\sum_{i=1}^{n}\,(e^{y_{i}{}^{ \lambda}}-1)(1+T_{i})}. \tag{10}\]  
**Theorem 1**. : _For a given \(\lambda>0\), the MLEs of \(\theta_{1}\) and \(\theta_{2}\) exist and are, respectively, given by \(\vec{\theta_{1}}\) and \(\vec{\theta_{2}}\) in Equation (10)._  
Proof. : Let \(t_{i}=\frac{\theta_{i}}{\theta_{i}},i=1,\,2\). Using inequality \(\ln t_{i}\leq t_{i}-1\), we obtain  
\[\ln\theta_{1}\leq\frac{\theta_{1}}{nk}\sum_{i=1}^{n}\sum_{j=1}^{k}(e^{x_{ij}{} ^{\lambda}}-1)(1+S_{j})-1+\ln\vec{\theta_{1}}, \tag{11}\]  
\[\ln\theta_{2}\leq\frac{\theta_{2}}{n}\sum_{i=1}^{n}(e^{y_{i}{}^{\lambda}}-1)( 1+T_{i})-1+\ln\vec{\theta_{2}}. \tag{12}\]  
By substituting Equations (11) and (12) into (7), one obtains  
\[ \begin{split} l(\theta_{1},\theta_{2},\lambda)\leq& n(k+1)\ln\lambda+nk\ln\vec{\theta_{1}}+n\ln\vec{\theta_{2}}-nk-n\\ &+\sum_{i=1}^{n}\sum_{j=1}^{k}\,[(\lambda-1)\ln x_{ij}+x_{ij}{}^ {\lambda}]\\ &+\sum_{i=1}^{n}\,[(\lambda-1)\ln y_{i}+y_{i}{}^{\lambda}].\end{split} \]  
From Equation (10), one obtains  
\[nk=\vec{\theta_{1}}\sum_{i=1}^{n}\sum_{j=1}^{k}\,(e^{x_{ij}{}^{\lambda}}-1)( 1+S_{j}),\,\,n=\vec{\theta_{2}}\sum_{i=1}^{n}\,(e^{y_{i}{}^{\lambda}}-1)(1+T_{ i}).\]  
One further obtains  
\[ \begin{split} l(\theta_{1},\theta_{2},\lambda)\leq& n(k+1)\ln\lambda+nk\ln\vec{\theta_{1}}+n\ln\vec{\theta_{2}}\\ &+\sum_{i=1}^{n}\sum_{j=1}^{k}\,[(\lambda-1)\ln x_{ij}+x_{ij}{}^ {\lambda}+\vec{\theta_{1}}(1-e^{x_{ij}{}^{\lambda}})(1+S_{j})]\\ &+\sum_{i=1}^{n}\,[(\lambda-1)\ln y_{i}+y_{i}{}^{\lambda}+\vec{ \theta_{2}}(1-e^{y_{i}{}^{\lambda}})(1+T_{i})]\\ &=l(\vec{\theta_{1}},\vec{\theta_{2}},\lambda).\end{split} \]  
The equality holds if \(t_{i}=1,\,\,i=1,\,2\), where \(\theta_{1}=\vec{\theta_{1}}\) and \(\theta_{2}=\vec{\theta_{2}}\). From Theorem 1, by substituting \(\theta_{1}=\vec{\theta_{1}}\) and \(\theta_{2}=\vec{\theta_{2}}\) into Equation (7), the log-likelihood function of \(\lambda\) is  
\[