# Optimal Confidence Regions for Weibull Parameters and Quantiles under Progressive Censoring
## 2 Weibull Models and Progressive Censoring  
Suppose that the lifetime \(X\) of a certain device follows the Weibull distribution with scale parameter \(\theta>0\) and shape parameter \(\alpha>0\), which is denoted as \(X\sim W(\theta,\alpha)\). The probability density function (pdf) and cumulative distribution function (cdf) of \(X\) are then given by  
\[f(x\mid\theta,\alpha)=\frac{\alpha x^{\alpha-1}/\theta^{\alpha}}{\exp\{\big{(} x/\theta\big{)}^{\alpha}\}}\quad\text{and}\quad F(x\mid\theta,\alpha)=1-\exp\Bigl{\{}- \Bigl{(}\frac{x}{\theta}\Bigr{)}^{\alpha}\Bigr{\}},\quad x>0, \tag{1}\]  
respectively. Moreover, the reliability or survival function of \(X\) is defined as \(S(x\mid\theta,\alpha)=\exp\bigl{\{}-(x/\theta)^{\alpha}\bigr{\}}\) and the failure rate is given by \(h(x\mid\theta,\alpha)=(\alpha/\theta^{\alpha})x^{\alpha-1}\) for \(x>0\). The \(k\)-th moment of \(X\sim W(\theta,\alpha)\) is obtained to be \(E[X^{k}\mid\theta,\alpha]=\theta^{k}\Gamma(1+k/\alpha)\), \(k=1,2,\dots\), where \(\Gamma(\cdot)\) is the well-known gamma function. The parameter \(\theta\) determines the scaling of the density, whereas the parameter \(\alpha\) controls its shape. In many practical cases, the survival of populations with increasing (\(\alpha>1\)), decreasing (\(\alpha<1\)), or constant (\(\alpha=1\)) hazard risks can be modeled by Weibull distributions. Assume now that \(n\) randomly selected units from a \(W(\theta,\alpha)\) population with unknown parameters \(\theta\) and \(\alpha\) are put on life test under a progressive censoring scheme \(\mathbf{r}=(r_{1},\dots,r_{s})\), and also that \(\mathbf{x}=(x_{1},\dots,x_{s})\) is the the observed realization of the random sample of failure times \(\mathbf{X}=(X_{1},\dots,X_{s})\). That is, \(n_{0}=n\) units are simultaneously placed on test at time zero in the life-testing experiment; for \(i=1,\dots,s-1\), \(r_{i}\) randomly selected living units are retired from the study at the \(i\)-th observed failure time, \(x_{i}\); so, prior to the \((i+1)\)-th failure, there are \(n_{i}=n-i-\sum_{j=1}^{i}r_{j}\) units on inspection; finally, at the time of the \(s\)-th observed failure, \(x_{s}\), the test is concluded, i.e., the remaining \(r_{s}\) units are removed from the analysis. The constants \(s\) and \(r_{i}\), \(i=1,\dots,s\), are prefixed integers which must satisfy the assumptions: \(1\leq s\leq n\), \(0\leq r_{i}\leq n_{i-1}-1\) for \(i=1,\dots,s-1\), and \(r_{s}=n_{s-1}-1\). The likelihood function for \((\theta,\alpha)\) given \((\mathbf{x},\mathbf{r})\), is then defined by  
\[L(\theta,\alpha\mid\mathbf{x},\mathbf{r})=\prod_{i=1}^{s}n_{i-1}f(x_{i}\mid \theta,\alpha)\{1-F(x_{i}\mid\theta,\alpha)\}^{r_{i}},\quad\theta,\alpha>0. \tag{2}\]  
In accordance with (1) and (2), the likelihood becomes  
\[L(\theta,\alpha\mid\mathbf{x},\mathbf{r})=\frac{Nw^{\alpha-1}a^{s}\theta^{-s \alpha}}{\exp(v_{\alpha}/\theta^{\alpha})},\quad\theta,\alpha>0,\]  
where \(w=W(\mathbf{x})\) and \(v_{\alpha}=V_{\alpha}(\mathbf{x},\mathbf{r})\) are the observed values of the random quantities  
\[W\equiv W(\mathbf{X})=\prod_{i=1}^{s}X_{i}\quad\text{and}\quad V_{\alpha} \equiv V_{\alpha}(\mathbf{X},\mathbf{r})=\sum_{i=1}^{s}(r_{i}+1)X_{i}^{ \alpha},\]  
respectively, and \(N=\prod_{i=1}^{s}n_{i-1}\). Given the censoring scheme \(\mathbf{r}\), it is clear that the observed sample \(\mathbf{X}\) is minimal sufficient for \((\theta,\alpha)\). Hereafter, it will be assumed that \(s>2\) and \(x_{1}<x_{s}\). Note that the probability that \(X_{1}\geq X_{s}\) is zero when \(s\geq 2\). In such a case, the unique MLE of \((\theta,\alpha)\), denoted by \((\widehat{\theta},\widehat{\alpha})\), can be derived by solving the equations \(\partial\log L(\theta,\alpha\mid\mathbf{x},\mathbf{r})/\partial\theta=0\) and \(\partial\log L(\theta,\alpha\mid\mathbf{x},\mathbf{r})/\partial\alpha=0\) via iterative procedures. In our situation, the MLE of \((\theta,\alpha)\) is an insufficient statistic.