## 2 Estimation of \(R_{s,k}\) by Maximum Likelihood  
The random variables \(X\) and \(Y\) are independently distributed. Each follows an exponentiated moment-based exponential distribution of parameters \((\alpha_{1}\), \(\beta)\) for \(X\) and \((\alpha_{2}\), \(\beta)\) for \(Y\), where \(\beta\) is the unknown common scale parameter and the shape parameters \(\alpha_{1}\,\text{and}\,\alpha_{2}\) are unknown. The multicomponent stress-strength reliability for the exponentiated moment-based exponential distribution using Equation (3) is as follows:  
\[ \begin{split} R_{s,k}&=\sum\limits_{i=s}^{k}\left( \begin{array}{c}k\\ i\end{array}\right)\int\limits_{0}^{1}(1-t^{\alpha_{1}})^{i}(t^{\alpha_{1}})^{ k-i}\alpha_{2}t^{\alpha_{2}-1}dt,\quad\text{where}\;\;t=1-\left(1+\frac{y}{ \beta}\right)e^{-\frac{y}{\beta}}\\ &=\sum\limits_{i=s}^{k}\left(\begin{array}{c}k\\ i\end{array}\right)\int\limits_{0}^{1}(1-z)^{i}(z)^{k-i}\alpha_{2}(z^{\frac{1 }{\alpha_{1}}})^{\alpha_{2}-1}\frac{z^{\frac{1}{\alpha_{1}-1}}}{\alpha_{1}}dz,\quad\text{where}\;\;z=t^{\alpha_{1}}\\ &\quad\quad=v\sum\limits_{i=s}^{k}\left(\begin{array}{c}k\\ i\end{array}\right)\int\limits_{0}^{1}(1-z)^{i}(z)^{k-i+v-1}dz,\quad\text{ where}\;\;v=\frac{\alpha_{2}}{\alpha_{1}}\\ &=v\sum\limits_{i=s}^{k}\left(\begin{array}{c}k\\ i\end{array}\right)\text{B}(i+1,k-i+v),\quad\text{ where}\;\text{B}(\text{m},\text{n})\;\text{is the beta first kind function.}\end{split}  \tag{4}\]  
After simplification, we obtain the following:  
\[R_{s,k}=v\sum\limits_{i=s}^{k}\frac{k!}{(k-i)!}\left(\prod\limits_{j=0}^{i} \left(k+v-j\right)\right)^{-1}, \tag{5}\]  
since \(k\) and \(i\) are integers and \(v=\frac{\alpha_{2}}{\alpha_{1}}\) can be obtained in Equations (11) and (12). Equation (5) is a multicomponent stress-strength reliability of the system. Let \(\left(X_{1},X_{2},\dots,X_{n}\right)\) be a random sample of strength variables and \(\left(Y_{1},Y_{2},\dots,Y_{m}\right)\) be a random sample of stress variables following an exponentiated moment-based exponential distribution with shape parameters \(\alpha_{1}\) and \(\alpha_{2}\) and common scale parameter \(\beta\). The log-likelihood associated with the observed sample is as follows:  
\[L(\alpha_{1},\alpha_{2},\beta)=\prod\limits_{i=1}^{n}f(x_{i},\alpha_{1}, \alpha_{2})\prod\limits_{j=1}^{m}f(y_{j},\alpha_{1},\alpha_{2}), \tag{6}\]  
\[ \begin{split}\ln L&=n\ln\alpha_{1}+m\ln\alpha_{2} -2(m+n)\ln\beta+\sum\limits_{i=1}^{n}\ln x_{i}+\sum\limits_{j=1}^{m}\ln y_{j} -\frac{1}{\beta}\sum\limits_{i=1}^{n}x_{i}-\frac{1}{\beta}\sum\limits_{j=1}^ {m}y_{j}\\ &\quad+(\alpha_{1}-1)\sum\limits_{i=1}^{n}\ln\left(1-\left(1+ \frac{x_{i}}{\beta}\right)e^{-\frac{x_{i}}{\beta}}\right)+(\alpha_{2}-1)\sum \limits_{j=1}^{m}\ln\left(1-\left(1+\frac{y_{i}}{\beta}\right)e^{-\frac{y_{i} }{\beta}}\right).\end{split}  \tag{7}\]  
The maximum likelihood estimators \(\hat{\alpha}_{1},\hat{\alpha}_{2},\;\text{and}\;\hat{\beta}\) are obtained iteratively as a solution of the following:  
\[\frac{\partial\ln\text{L}}{\partial\alpha_{1}}=0\;\;\Rightarrow\;\;\frac{n} {\alpha_{1}}+\sum\limits_{i=1}^{n}\ln\left(1-\left(1+\frac{x_{i}}{\beta} \right)e^{-\frac{x_{i}}{\beta}}\right)=0\;\;\Rightarrow\;\;\hat{\alpha}_{1}= \frac{-n}{\sum\limits_{i=1}^{n}\ln\left(1-\left(1+\frac{x_{i}}{\beta}\right)e^ {-\frac{x_{i}}{\beta}}\right)}, \tag{8}\]  
\[ \begin{split}\frac{\partial\ln\text{L}}{\partial\alpha_{2}}=0\;\; \Rightarrow\;\;\frac{m}{\alpha_{2}}+\sum\limits_{j=1}^{m}\ln\left(1-\left(1+ \frac{y_{i}}{\beta}\right)e^{-\frac{y_{i}}{\beta}}\right)=0\;\;\Rightarrow\;\; \hat{\alpha}_{2}=\frac{-m}{\sum\limits_{j=1}^{m}\ln\left(1-\left(1+\frac{y_{ i}}{\beta}\right)e^{-\frac{y_{i}}{\beta}}\right)},\end{split}  \tag{9}\]  
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