## 6 Maximum Likelihood Estimation (MLE)  
Let \(\chi_{1},\chi_{2},\ldots,\chi_{n}\) are observed values of \(X_{1:n},X_{2:n},\ldots X_{n:n}\) that is an ordered random sample from the E-WD. The E-WD likelihood is  
\[l=\left(\frac{\psi\lambda\theta}{e^{\psi}-1}\right)^{n}\left(\prod_{i=1}^{n} \chi_{i}^{-1+\theta}\right)e^{\sum_{i=1}^{n}\left(e^{-\chi_{i}^{\theta}\lambda _{i}}\psi-\chi_{i}^{\theta}\lambda\right)}\, \tag{38}\]  
and the corresponding log-likelihood function \((L)\) is  
\[L=n\text{Log}\left[\frac{\psi\lambda\theta}{e^{\psi}-1}\right]+\sum_{i=1}^{n} \text{Log}\left[\chi_{i}^{-1+\theta}\right]+\sum_{i=1}^{n}\left(e^{-\chi_{i}^ {\theta}\lambda}\psi-\chi_{i}^{\theta}\delta_{1}\right), \tag{39}\]  
Taking the first partial derivatives of log-likelihood (39) with respect to \(\psi,\lambda,\theta\) and equating each to zero. \[\frac{\partial}{\partial\psi}L = \frac{n\left(e^{\psi}-1\right)}{\psi\lambda\theta}\left(\frac{ \delta_{2}\lambda}{e^{\psi}-1}-\frac{e^{\psi}\psi\lambda\theta}{\left(e^{\psi }-1\right)^{2}}\right)+\sum_{i=1}^{n}e^{-\lambda\lambda_{i}^{\theta}}=0, \tag{40}\] \[\frac{\partial}{\partial\theta}L = \frac{n}{\theta}+\sum_{i=1}^{n}\text{Log}[\chi_{i}]+\sum_{i=1}^{n }\left(-\lambda\text{Log}[\chi_{i}]\chi_{i}^{\theta}-e^{-\lambda\lambda_{i}^ {\theta}}\psi\lambda\text{Log}[\chi_{i}]\chi_{i}^{\theta_{2}}\right)=0,\] (41) \[\frac{\partial}{\partial\lambda}L = \frac{n}{\lambda}+\sum_{i=1}^{n}\left(-\chi_{i}^{\theta}-e^{- \lambda\lambda_{i}^{\theta}}\psi\chi_{i}^{\theta}\right)=0. \tag{42}\]  
Solving Equation (42) for \(\psi\), we have  
\[\psi=\frac{\frac{n}{\lambda}-\sum_{i=1}^{n}\chi_{i}^{\theta}}{\sum_{i=1}^{n}e ^{-\lambda\lambda_{i}^{\theta}}\chi_{i}^{\theta}}. \tag{43}\]  
Solving Equations (40) and (41) after substituting Equation (43), we get the maximum likelihood estimators \(\hat{\psi}_{ML}\), \(\hat{\lambda}_{ML}\), \(\hat{\theta}_{ML}\) of the \(E-WD\left(\psi,\lambda,\theta\right)\) parameters. The asymptotic confidence intervals of the parameters \(\psi\), \(\lambda\) and \(\theta\). Then \(\hat{V}=V(\hat{\psi}_{ML}\), \(\hat{\lambda}_{ML}\), \(\hat{\theta}_{ML})=[\sigma_{i,j}],i,j=1\), 2, 3 is the observed variance covariance matrix, such that  
\[V(\psi,\lambda,\theta)=-\left[ \begin{array}{ccc}\frac{\partial^{2}L}{ \partial\psi^{2}}&\frac{\partial^{2}L}{\partial\psi\partial\lambda}&\frac{ \partial^{2}L}{\partial\psi\partial\theta}\\ \\ \frac{\partial^{2}L}{\partial\lambda\partial\psi}&\frac{\partial^{2}L}{ \partial\lambda^{2}}&\frac{\partial^{2}L}{\partial\lambda\partial\theta}\\ \\ \frac{\partial^{2}L}{\partial\theta\partial\psi}&\frac{\partial^{2}L}{ \partial\theta\partial\lambda}&\frac{\partial^{2}L}{\partial\theta^{2}}\\ \end{array} \right]^{-1}, \tag{44}\]  
A \(100(1-\epsilon)\%\) two-sided approximate confidence intervals for the parameters \(\psi\), \(\lambda\), and \(\theta\) are then given by  
\[\widehat{\psi}\pm z_{\epsilon/2}\sqrt{V(\widehat{\psi})}, \tag{45}\]  
\[\widehat{\lambda}\pm z_{\epsilon/2}\sqrt{V(\widehat{\lambda})}, \tag{46}\]  
and  
\[\widehat{\theta}\pm z_{\epsilon/2}\sqrt{V(\widehat{\theta})}, \tag{47}\]respectively, where \(V(\hat{\psi})\), \(V(\hat{\lambda})\), and\(V(\hat{\theta})\), are the estimated variances of \(\hat{\psi}_{ML}\), \(\hat{\lambda}_{ML}\), and \(\hat{\theta}_{ML}\), which are given by the diagonal elements of \(\hat{V}\), and \(z_{\epsilon/2}\) is the upper \(\left(\frac{\epsilon}{2}\right)\) percentile of the standard normal distribution. Next, obtain the bootstrap confidence intervals for boot-p for the unknown parameters \(\delta=(\psi,\,\lambda,\,\theta)\), we apply the following algorithms  
1. Generate sample \(\{\chi_{i}\}\) of size n from the \(E-WD\left(\psi,\,\lambda,\,\theta\right)\) and estimate a \(\hat{\delta}\). 2. Generate another sample \(\{\chi_{i}^{*}\}\) of size n using \(\hat{\delta}\). Then estimate \(\hat{\delta}^{*}\). 3. Repeat step 2 B times. 4. Via \(\hat{F}(x)=P(\hat{\delta}^{*}\leq x)\), that is, the CDF of \(\hat{\delta}^{*}\), the \(100(1-\epsilon)\%\) C.I.