# Optimal Confidence Regions for Weibull Parameters and Quantiles under Progressive Censoring
## 4 Optimal Regions for Two Weibull Quantiles  
Given \(0<u<1\), the Weibull \(u\)-quantile is defined as \(q_{u}=\theta\{-\log(1-u)\}^{1/\alpha}\). Our goal in this section is to construct the smallest-size confidence region for the pair of Weibull quantiles \((q_{a},q_{b})\), where \(0<a<b<1\). The posterior pdf of \((q_{a},q_{b})\) given \((\mathbf{x},\mathbf{r})\) can be expressed as  
\[\pi^{*}_{a,b}(q_{a},q_{b}\mid\mathbf{x},\mathbf{r})=\pi^{*}(\theta,\alpha\mid \mathbf{x},\mathbf{r})\bigg{|}\frac{\partial(\theta,\alpha)}{\partial(q_{a},q _{b})}\bigg{|},\quad 0<q_{a}<q_{b},\]  
where \(\partial(\theta,\alpha)/\partial(q_{a},q_{b})\) denotes the Jacobian (determinant) for the change of variables from \((\theta,\alpha)\) to \((q_{a},q_{b})\). The Jacobian is defined by  
\[\frac{\partial(\theta,\alpha)}{\partial(q_{a},q_{b})}=\left| \begin{array}{ cc}\frac{\partial\theta}{\partial q_{a}}&\frac{\partial\theta}{\partial q_{b}}\\ \frac{\partial\alpha}{\partial q_{a}}&\frac{\partial\alpha}{\partial q_{b}} \end{array} \right|=\frac{\partial\theta}{\partial q_{a}}\frac{\partial\alpha} {\partial q_{b}}-\frac{\partial\theta}{\partial q_{b}}\frac{\partial\alpha}{ \partial q_{a}},\]  
where \(q_{a}=\theta\{-\log(1-a)\}^{1/\alpha}\) and \(q_{b}=\theta\{-\log(1-b)\}^{1/\alpha}\). After some calculations, it is derived that  
\[\frac{\partial(\theta,\alpha)}{\partial(q_{a},q_{b})}=\frac{(k_{a}-k_{b})q_{b }^{k_{b}/(k_{a}-k_{b})}}{q_{a}^{k_{a}/(k_{a}-k_{b})}\{\log(q_{a}/q_{b})\}^{2}},\]  
where \(k_{j}=\log\{-\log(1-j)\}\) for \(j=a,b\), because  
\[\frac{\partial\theta}{\partial q_{a}}=\frac{-k_{b}}{k_{a}-k_{b}}\bigg{(}\frac{ q_{b}}{q_{a}}\bigg{)}^{k_{a}/(k_{a}-k_{b})},\quad\frac{\partial\theta}{ \partial q_{b}}=\frac{k_{a}}{k_{a}-k_{b}}\bigg{(}\frac{q_{b}}{q_{a}}\bigg{)}^ {k_{b}/(k_{a}-k_{b})}\]  
and  
\[\frac{\partial\alpha}{\partial q_{a}}=\frac{k_{b}-k_{a}}{q_{a}\{\log(q_{a}/q_{ b})\}^{2}},\quad\frac{\partial\alpha}{\partial q_{b}}=\frac{k_{a}-k_{b}}{q_{b}\{ \log(q_{a}/q_{b})\}^{2}}.\]  
As a consequence of the above results, the posterior pdf of \((q_{a},q_{b})\) given \((\mathbf{x},\mathbf{r})\) is defined as  
\[\pi^{*}_{a,b}(q_{a},q_{b}\mid\mathbf{x},\mathbf{r})=\pi^{*}\Bigg{(}\frac{q_{b }^{k_{a}/(k_{a}-k_{b})}}{q_{a}^{k_{b}/(k_{a}-k_{b})}},\frac{k_{a}-k_{b}}{\log (q_{a}/q_{b})}\mid\mathbf{x},\mathbf{r}\Bigg{)}\frac{(k_{b}-k_{a})q_{b}^{k_{b} /(k_{a}-k_{b})}}{q_{a}^{k_{a}/(k_{a}-k_{b})}\{\log(q_{a}/q_{b})\}^{2}}\]  
for \(0<q_{a}<q_{b}\). Obviously, \(\big{\{}(q_{a;i},q_{b;i})\big{\}}_{i=1}^{m}\), where  
\[q_{a;i}=\theta_{i}\{-\log(1-a)\}^{1/\alpha_{i}}\quad\text{and}\quad q_{b;i}= \theta_{i}\{-\log(1-b)\}^{1/\alpha_{i}}\]  
for \(i=1,\ldots,m\), constitute a random sample of size \(m\) from the posterior distribution of \((q_{a},q_{b})\) conditional to \((\mathbf{x},\mathbf{r})\). In this case, the minimum-area \(\varepsilon\)-confidence region for \((q_{a},q_{b})\), denoted by \(R_{a,b;x}\equiv R_{a,b;x}(\mathbf{x},\mathbf{r})\), is defined as  
\[R_{a,b;x}=\Big{\{}(q_{a},q_{b}):\pi^{*}_{a,b}(q_{a},q_{b}\mid\mathbf{x},\mathbf{ r})\geq d_{\varepsilon}\Big{\}},\]  
where  
\[\Pr\big{\{}(q_{a},q_{b})\in R_{a,b;x}\mid\mathbf{x},\mathbf{r}\big{\}}=\iint_{ R_{a,b;x}}\pi^{*}_{a,b}(q_{a},q_{b}\mid\mathbf{x},\mathbf{r})\mathrm{d}q_{a} \mathrm{d}q_{b}=\varepsilon.\]  
As discussed, the smallest-size confidence regions are relevant to practitioners because they are less likely to contain spurious parameter values. An approximation of the constant \(d_{\varepsilon}\) is given by the \((1-\varepsilon)\)-quantile of the simulated sample  
\[\Big{\{}\pi^{*}_{a,b}(q_{a;i},q_{b;i}\mid\mathbf{x},\mathbf{r})\Big{\}}_{i=1 }^{m}\]  
because \(d_{\varepsilon}\) is precisely the \((1-\varepsilon)\)-quantile of the random variable \(\pi^{*}_{a,b}(q_{a},q_{b}\mid\mathbf{x},\mathbf{r})\).