# Optimal Confidence Regions for Weibull Parameters and Quantiles under Progressive Censoring
## 3 Optimal Regions for the Weibull Parameters  
Assume that \(\varepsilon\in(0,1)\) denotes the confidence or credibility level and also that \(c_{\varepsilon}\) represents the \((1-\varepsilon)\)-quantile of the random posterior density of \((\theta,\alpha)\) given \((\mathbf{x},\mathbf{r})\). The Bayesian HPD \(\varepsilon\)-credibility region for \((\theta,\alpha)\) is then defined as  
\[R_{\varepsilon}\equiv R_{\varepsilon}(\mathbf{x},\mathbf{r})=\{(\theta, \alpha):\pi^{*}(\theta,\alpha\mid\mathbf{x},\mathbf{r})\geq c_{\varepsilon}\},\]  
where  
\[\Pr\{(\theta,\alpha)\in R_{\varepsilon}\mid\mathbf{x},\mathbf{r}\}=\iint_{R_ {\varepsilon}}\pi^{*}(\theta,\alpha\mid\mathbf{x},\mathbf{r})\mathrm{d} \theta\mathrm{d}\alpha=\varepsilon.\]  
The region \(R_{\varepsilon}\) is also the conditional (frequentist) \(\varepsilon\)-confidence region for \((\theta,\alpha)\) with minimum area. The Bayesian credibility degree \(\varepsilon\) coincides with the frequency-based confidence level of the random region \(R_{\varepsilon}(\mathbf{X},\mathbf{r})\). Therefore, the smallest-size \(100\varepsilon\%\) confidence set for \((\theta,\alpha)\) based on the Conditionality Principle given \((\mathbf{x},\mathbf{r})\) may be defined as  
\[R_{\varepsilon}\equiv R_{\varepsilon}(\mathbf{x},\mathbf{r})=\Big{\{}(\theta, \alpha):w^{\alpha-1}\theta^{-s\alpha-1}\alpha^{s-1}\exp(-v_{\alpha}/\theta^{ \alpha})\geq Kc_{\varepsilon}\Big{\}}.\]  
Since the joint posterior pdf of \(\theta\) and \(\alpha\) derived in (3) is unimodal, it is clear that \(R_{\varepsilon}\) is a simply connected region. Hence, the set \(R_{\varepsilon}\) is bounded by a single curve \(C_{\varepsilon}\), which does not intersect itself, i.e., the region limited by the contour \(C_{\varepsilon}\) results in the required \(\varepsilon\)-confidence set. An approximate value of \(c_{\varepsilon}\) can be obtained through simulation. A simple algorithm for determining a random sample of size \(m\) from the posterior distribution of \((\theta,\alpha)\) conditional to \((\mathbf{x},\mathbf{r})\) can be sketched as follows: Given a large integer number \(m\), for \(i=1,\ldots,m\), simulate an observation \(u_{i}\) from the uniform distribution \(U(0,1)\) and another value \(d_{i}\) from the \(\chi^{2}_{2S}\) distribution, and then determine \(\alpha_{i}=G^{-1}(u_{i})\) and \(\theta_{i}=(2v_{\alpha_{i}}/d_{i})^{1/\alpha_{i}}\). In such a case, \((\theta_{1},\alpha_{1}),\ldots,(\theta_{m},\alpha_{m})\) constitute a random sample of size \(m\) from the posterior distribution of \((\theta,\alpha)\). Since \(c_{\varepsilon}\) is the \((1-\varepsilon)\)-quantile of the random variable \(\pi^{*}(\theta,\alpha\mid\mathbf{x},\mathbf{r})\), an approximation of \(c_{\varepsilon}\) is given by the \((1-\varepsilon)\)-quantile of the simulated sample \(\{\pi^{*}(\theta_{i},\alpha_{i}\mid\mathbf{x},\mathbf{r})\}_{i=1}^{m}\). For interested readers, [PERSON] [38] focuses on the fundamentals of Monte Carlo methods using basic computer simulation techniques. The smallest confidence regions presented in this paper can also be applied in hypotheses testing. For instance, if \(\mathbf{x}\) is the observed value of \(\mathbf{X}\) and \(\mathbf{r}\) is the progressive censoring scheme, the \(p\)-value associated to the test of the null hypothesis \(H_{0}:(\theta,\alpha)=(\theta_{0},\alpha_{0})\) versus the alternative hypothesis \(H_{1}:(\theta,\alpha)\
eq(\theta_{0},\alpha_{0})\) based on the smallest confidence sets for \((\theta,\alpha)\) would be defined by \(p=1-\varepsilon_{0}\), where  
\[\varepsilon_{0}=\min\{\varepsilon:(\theta_{0},\alpha_{0})\in R_{\varepsilon }\}.\]  
Figure 3: Posterior pdf of \(\alpha\) in Section 5. It is easy to show that  
\[R_{\varepsilon_{0}}=\{(\theta,\alpha):\pi^{*}(\theta,\alpha\mid\mathbf{x}, \mathbf{r})\geq\pi^{*}(\theta_{0},\alpha_{0}\mid\mathbf{x},\mathbf{r})\},\]  
i.e., the constant \(\varepsilon_{\varepsilon_{0}}\) equals \(\pi^{*}(\theta_{0},\alpha_{0}\mid\mathbf{x},\mathbf{r})\).