## 5 Estimation of the UEHLD's Parameters
### ML Estimators  
Assume \(t_{1},\ldots,t_{n}\) are the observed values from the UEHLD with parameters \(\varphi\), and \(\delta\). The likelihood function, say \(L(t|\varphi,\delta)\), of the UEHLD is expressed as:  
\[L(t|\varphi,\delta)=(2\varphi\delta)^{n}\prod_{l=1}^{n}\frac{t_{i}^{\delta-1}} {\left(1+t_{i}^{\delta}\right)^{2}}\,\left(\frac{1-t_{i}^{\delta}}{1+t_{i}^{ \delta}}\right)^{\varphi-1}. \tag{14}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\xi\)} & \multirow{2}{*}{**Measures**} & \multirow{2}{*}{\(\varphi=0.5\), \(\delta=0.5\)} & \multirow{2}{*}{\(\varphi=0.5\), \(\delta=1\)} & \multirow{2}{*}{\(\varphi=0.5\)} & \multirow{2}{*}{\(\varphi=1\), \(\delta=0.5\)} & \multirow{2}{*}{\(\varphi=1\), \(\delta=1.5\)} & \multirow{2}{*}{\(\varphi=1\), \(\delta=1.5\)} & \multirow{2}{*}{\(\delta=0.5\)} & \multirow{2}{*}{\(\delta=0.5\)} & \multirow{2}{*}{\(\delta=1\)} & \multirow{2}{*}{\(\delta=1.5\)} \\ \cline{3-3} \cline{5-10}  & & \(\mathbb{R}(\zeta)\) & \(-0.229\) & \(-0.098\) & \(-0.185\) & \(-0.56\) & \(-1.022\) & \(-0.064\) & \(-0.099\) \\ \cline{2-10}  & & \(\hbar(\zeta)\) & \(-0.374\) & \(-0.162\) & \(-0.303\) & \(-0.878\) & \(-1.516\) & \(-0.106\) & \(-0.165\) \\ \cline{2-10}  & & \(\aleph(\zeta)\) & \(-0.224\) & \(-0.097\) & \(-0.181\) & \(-0.53\) & \(-0.924\) & \(-0.063\) & \(-0.098\) \\ \hline \multirow{4}{*}{\(1.5\)} & \(\mathbb{R}(\zeta)\) & \(-0.682\) & \(-0.365\) & \(-0.565\) & \(-1.557\) & \(-2.295\) & \(-0.118\) & \(-0.154\) \\ \cline{2-10}  & & \(\hbar(\zeta)\) & \(-0.872\) & \(-0.441\) & \(-0.708\) & \(-2.323\) & \(-3.924\) & \(-0.137\Then, the log likelihood function, say \(\ell_{1}\), of the UEHLD is given as  
\[\ell_{1} =n[\ln(2)+\ln(\varphi)+\ln(\delta)]+(\delta-1)\sum\limits_{i=1}^{n} \ln(t_{i})-2\sum\limits_{i=1}^{n}\ln\bigl{(}1+t_{i}^{\delta}\bigr{)}\] \[+(\varphi-1)\sum\limits_{i=1}^{n}\left[\ln\bigl{(}1-{t_{i}}^{ \delta}\bigr{)}-\ln\bigl{(}1+t_{i}^{\delta}\bigr{)}\right].\]  
Therefore, the ML equations are given by:  
\[\frac{\partial\ell_{1}}{\partial\varphi}=\frac{n}{\varphi}+\sum \limits_{i=1}^{n}\left[\ln\bigl{(}1-{t_{i}}^{\delta}\bigr{)}-\ln\bigl{(}1+t_{ i}^{\delta}\bigr{)}\right],\] \[\frac{\partial\ell_{1}}{\partial\delta}=\frac{n}{\delta}+\sum \limits_{i=1}^{n}\ln\bigl{(}t_{i}\bigr{)}-2\sum\limits_{i=1}^{n}\frac{t_{i}^{ \delta}\ln(t_{i})}{1+t_{i}^{\delta}}-(\varphi-1)\sum\limits_{i=1}^{n}\Bigl{[} \frac{2t_{i}^{\delta}\ln(t_{i})}{1-t_{i}^{\delta}}\Bigr{]}.\]  
Solving the non-linear equations \(\partial\ell_{1}/\partial\varphi=0\), and \(\partial\ell_{1}/\partial\delta\) numerically by using optimization algorithms such as the Newton-Raphson (NR) algorithm, we determine the ML estimators of \(\varphi\), and \(\delta\).  
Furthermore, it is known that, under regularity conditions, the asymptotic distribution of the ML estimators of the UEHLD parameters is given by:  
\[(\phi-\varphi),(\delta-\delta)\sim N(0,I^{-1}(\varphi,\delta)),\]  
where \(I^{-1}(\varphi,\delta)\) is the variance-covariance matrix of the UEHLD parameters. Therefore, the two-sided approximate \((1-\alpha)\%\) CIs for the ML estimates of \(\varphi,\delta\) can be obtained as follows:  
\[L_{\varphi}=\phi-z_{\alpha/2}\sqrt{\mathrm{var}(\phi)}\,L_{ \varphi}=\phi+z_{\alpha/2}\sqrt{\mathrm{var}(\phi)},\] \[\mathrm{and}\] \[L_{\delta}=\delta-z_{\alpha/2}\sqrt{\mathrm{var}(\delta)}\,L_{ \delta}=\delta+z_{\alpha/2}\sqrt{\mathrm{var}(\delta)}.\]  
where \(z_{\alpha/2}\) is the \(100(1-\alpha)\%\) th standard normal percentile and \(\mathrm{var}(.)\) denotes the diagonal elements of the variance-covariance matrix corresponding to the model parameters.