## 4 Approximate Maximum Likelihood Estimator of \(R\)  
The MLEs do not take explicit forms; therefore, we approximate the likelihood equation and derive explicit estimators of the parameters.  
Since the random variable \(X\) follows \(IW(\alpha,\theta)\), then \(V=\ln X\) has the extreme value distribution with pdf as  
\[f(v;\mu,\sigma)=-\frac{1}{\sigma}e^{\frac{v-\mu}{\sigma}-e^{\frac{v-\mu}{\sigma} }},\ -\infty<v<+\infty, \tag{15}\]  
where \(\mu=-\frac{1}{\alpha}\ln\theta\) and \(\sigma=-\frac{1}{\alpha}\). The \(\mu\) and \(\sigma\) are location and scale parameters, respectively. The pdf and cdf of the standard extreme value distribution can be obtained as  
\[g(v)=e^{-v-\sigma^{-v}},\ G(v)=e^{-e^{-v}}. \tag{16}\]  
Suppose \(X_{(1)}<X_{(2)}<...<X_{(n)}\) and \(Y_{(1)}<Y_{(2)}<...<Y_{(m)}\) are the ordered \(X_{i}\)s and \(Y_{j}\)s, we assume the following notations: \(T_{(i)}=\ln X_{(i)}\), \(Z_{(i)}=\frac{T_{(i)}-\mu_{1}}{-\sigma},\ i=1,...,n\) and \(S_{(j)}=\ln Y_{(j)}\), \(W_{(j)}=\frac{S_{(j)}-\mu_{2}}{-\sigma},\ j=1,...,m\), where \(\mu_{1}=-\frac{1}{\alpha}\ln\theta_{1}\), \(\mu_{2}=-\frac{1}{\alpha}\ln\theta_{2}\) and \(\sigma=-\frac{1}{\alpha}\).  
The log-likelihood function of the data \(T_{(1)},...,T_{(n)}\) and \(S_{(1)},...,S_{(m)}\) is  
\[L(\mu_{1},\mu_{2},\sigma)\propto-(m+n)\ln(-\sigma)+\sum_{i=1}^{n}\ln g(z_{(i) })+\sum_{j=1}^{m}\ln g(w_{(j)}). \tag{17}\]  
Differentiating (17) in regard to \(\mu_{1},\ \mu_{2}\)_and_\(\sigma\), the score equations are obtained as  
\[\frac{\partial L}{\partial\mu_{1}} =\frac{1}{\sigma}\sum_{i=1}^{n}\frac{g^{\prime}(z_{(i)})}{g(z_{(i )})}=0, \tag{18}\] \[\frac{\partial L}{\partial\mu_{2}} =\frac{1}{\sigma}\sum_{j=1}^{m}\frac{g^{\prime}(w_{(j)})}{g(w_{(j )})}=0,\] (19) \[\frac{\partial L}{\partial\sigma} =-(m+n)\frac{1}{\sigma}-\frac{1}{\sigma}\sum_{i=1}^{n}\frac{g^{ \prime}(z_{(i)})}{g(z_{(i)})}z_{(i)}-\frac{1}{\sigma}\sum_{j=1}^{m}\frac{g^{ \prime}(w_{(j)})}{g(w_{(j)})}w_{(j)}=0. \tag{20}\]  
We note that the function \(h(z_{(i)})=\frac{g^{\prime}(z_{(i)})}{g(z_{(i)})}\) makes the score Equation (18) nonlinear and intricate. Thus, we approximate the function \(h(z_{(i)})=\frac{g^{\prime}(z_{(i)})}{g(z_{(i)})}\) by expanding it in a Taylor series around \(c_{i}\ =\ E(Z_{(i)})\). Furthermore, we also approximate the function \(h(w_{(j)})=\frac{g^{\prime}(w_{(j)})}{g(w_{(j)})}\) by expanding it in a Taylor series around \(d_{j}=E(W_{(j)})\). From [18], it is known that  
\[G(Z_{(i)})\stackrel{{ d}}{{=}}U_{(i)},\]  
where \(U_{(i)}\) is the \(i\)-th order statistic from the uniform \(U(0,1)\) distribution. Therefore,  
\[Z_{(i)}\stackrel{{ d}}{{=}}G^{-1}(U_{(i)}),\]  
and  
\[c_{i}=EZ_{(i)}\approx G^{-1}(EU_{(i)})=G^{-1}(i/(n+1)).\]  
We use the following notations, \(p_{i}=\frac{i}{n+1}\), \(\overline{p}_{j}=\frac{j}{m+1}\); therefore, \(c_{i}=G^{-1}(p_{i})=-\ln(-\ln p_{i})\), \(d_{j}=G^{-1}(\overline{p}_{j})=-\ln(-\ln\overline{p}_{j})\).  
Once \(\tilde{\sigma}\) is obtained, \(\tilde{\mu_{1}}\) and \(\tilde{\mu_{2}}\) can be derived immediately. Hence, the AMLE of \(R\) is given by  
\[\tilde{R}=\frac{\tilde{\theta}_{2}}{\tilde{\theta}_{1}+\tilde{ \theta}_{2}}, \tag{29}\]  
where  
\[\tilde{\alpha}=-\frac{1}{\tilde{\sigma}},\ \tilde{\theta_{1}}=e^{\frac{1}{ \tilde{\sigma}}(A_{1}+B_{1}\tilde{\sigma})},\ \tilde{\theta_{2}}=e^{\frac{1}{\tilde{\sigma}}(A_{2}+B_{2}\tilde{ \sigma})}.\]