## 4 Parameter Estimation  
_Parameter Search for Two-Parameter EEP Distributions_  
Suppose that \(\mathbf{t}=(t_{1},t_{2},\ldots,t_{n})\) is an independent identically distributed (i.i.d.) EEP sample with parameters \(\theta\) and \(\eta\). Without loss of generality, assume \(t_{1}<t_{2}<\ldots<t_{n}\)  
Figure 5: The median residual lifetime function plot of EEP distribution with different parameters. Assume that there exists a positive integer \(m\) such that \(t_{m}<\theta^{\frac{1}{\eta}}<t_{m+1}\). The corresponding log-likelihood of \(t\) can be written as follows:  
\[ \begin{split} l(\theta,\eta;t)&=n\text{ln}(c)+n\text{ ln}(\eta)+m\text{ln}(\alpha+1)\\ &+(n-m)\text{ln}(\alpha)+(\eta-1)\sum_{i=1}^{m}\text{ln}(t_{i}) \\ &-(\alpha\eta+1)\sum_{j=m+1}^{n}\text{ln}(t_{j})+\alpha n\text{ln}( \theta)\\ &-[(\alpha+1)m]\text{ln}(\theta)-\frac{\alpha+1}{\theta}\sum_{i=1 }^{m}t_{i}^{\eta}\end{split}  \tag{20}\]  
Due to the existence of the location parameters (\(\theta\) and \(\eta\)) in the EEP model, the maximum likelihood estimates (MLE) of the model parameters cannot be obtained directly by maximizing the log-likelihood function \(l(\theta,\eta;t)\). [PERSON] and [PERSON] [13,14] introduced a grid-search algorithm to find the estimates of the model parameters. In fact, the estimates can also be obtained by utilizing the 'optim' function in R. The steps for finding MLE of the model parameters are as follows:  
1. Arrange the observations in the sample in increasing order such that \(t_{(1)}\leq t_{(2)}\leq\ldots\leq t_{(n)}\). 2. For \(m=1,2,\ldots,n-1\), maximize the objective function \(l(\theta,\eta;t)\) and obtain \((\hat{\theta}_{1},\hat{\eta}_{1})\), \((\hat{\theta}_{  
\[I(\Theta) = \begin{pmatrix}-\frac{\partial^{2}I(\theta,\eta\mathcal{I})}{\partial \theta^{2}}&-\frac{\partial^{2}I(\theta,\eta)\mathcal{I}}{\partial\theta \partial\eta}\\ -\frac{\partial^{2}I(\theta,\eta\mathcal{I})}{\partial\theta\partial\eta}&- \frac{\partial^{2}I(\theta,\eta\mathcal{I})}{\partial\eta^{2}}\\ \end{pmatrix} \] \[= \begin{pmatrix}I_{11}&I_{12}\\ I_{21}&I_{22}\end{pmatrix} ,\]  
where  
\[ \begin{cases}I_{11}=\frac{\alpha n-(a+1)m}{\theta^{2}}+\frac{2(a+1)}{\theta^{3 }}\sum_{i=1}^{m}t_{i}^{\eta}\\ I_{12}=-\frac{a+1}{\theta^{2}}\sum_{i=1}^{m}t_{i}^{\eta}\ln(t_{i})\\ I_{21}=-\frac{a+1}{\theta^{2}}\sum_{i=1}^{m}t_{i}^{\eta}\ln(t_{i})\\ I_{22}=\frac{\eta}{\eta^{2}}+\frac{a+1}{\theta^{2}}\sum_{i=1}^{m}t_{i}^{\eta}[ \ln(t_{i})]^{2}.\end{cases} \]  
Essentially, the expected value \(\mathrm{E}(I(\theta))\) is the Fisher information matrix \(\mathcal{I}(\Theta)\). Consider \(\hat{\Theta}=(\hat{\theta},\hat{\eta})\) as the MLE for \(\Theta=(\theta,\eta)\). By the asymptotic property of MLE, we have the following under appropriate regularity conditions:  
\[(\hat{\Theta}-\Theta)\overset{\mathcal{D}}{\rightarrow}\mathcal{N}(\theta_{ 2},\mathcal{I}^{-1}(\Theta)),\text{as }n\rightarrow\infty,\]  
where \(\mathcal{I}(\Theta)\) stands for the Fisher information matrix of \(\Theta\), \(\mathcal{D}\) represents convergence in distribution, and \(\mathcal{N}(\theta_{2},\mathcal{I}^{-1}(\Theta))\) is the bivariate normal distribution with covariance matrix \(\mathcal{I}^{-1}(\Theta)\). However, in reality, \(\mathcal{I}(\Theta)=\mathrm{E}(I(\theta))\) cannot be easily evaluated. Therefore, the observed Fisher information matrix \(\mathcal{I}(\hat{\Theta})\) is used as a substitute. \[\mathcal{I}(\hat{\Theta})= \begin{pmatrix}\hat{\mathcal{I}}_{11}&\hat{ \mathcal{I}}_{12}\\ \hat{\mathcal{I}}_{21}&\hat{\mathcal{I}}_{22}\end{pmatrix} ,\]  
where,  
\[ \begin{cases}\hat{\mathcal{I}}_{11}=\frac{\alpha n-(a+1)m}{\theta^{2}}+\frac{ 2(a+1)}{\theta^{3}}\sum_{i=1}^{m}t_{i}^{\hat{\eta}}\\ \hat{\mathcal{I}}_{12}=-\frac{a+1}{\theta^{2}}\sum_{i=1}^{m}t_{i}^{\hat{\eta} }\ln(t_{i})\\ \hat{\mathcal{I}}_{21}=-\frac{a+1}{\theta^{2}}\sum_{i=1}^{m}t_{i}^{\hat{\eta} }\ln(t_{i})\\ \hat{\mathcal{I}}_{22}=\frac{n}{\theta^{2}}+\frac{a+1}{\theta}\sum_{i=1}^{m}t_ {i}^{\hat{\eta}}[\ln(t_{i})]^{2}.\end{cases}