# Statistical Power Analysis in Reliability Demonstration Testing: The Probability of Test Success
## 3 Calculation of the Probability of Test Success
### Approximate Calculation for Failure-Based Tests
The likelihood can be extended for left-censored or truncated as well as interval-censored failure times. The variance-covariance matrix \(\mathbf{V}\) using (28) is the inverse of the Fisher information matrix, \(\mathbf{I}\)[8]. Because \(\mathbf{I}\) is symmetric and positive definite, its inverse and the matrix \(\mathbf{V}\) can be calculated using  
\[\mathbf{V} = \begin{bmatrix}\text{Var}(T)&\text{Cov}(T,b)\\ \text{Cov}(b,T)&\text{Var}(b)\end{bmatrix} =\mathbf{I}^{-1}= \begin{bmatrix}- \frac{\partial^{2}\Lambda}{\partial T^{2}}&-\frac{\partial^{2}\Lambda}{ \partial T\partial b}\\ -\frac{\partial^{2}\Lambda}{\partial b\partial T}&-\frac{\partial^{2}\Lambda}{ \partial b^{2}}\end{bmatrix} ^{-1}=\frac{1}{\det(\mathbf{I})} \begin{bmatrix}- \frac{\partial^{2}\Lambda}{\partial b^{2}}&\frac{\partial^{2}\Lambda}{ \partial T\partial b}\\ \frac{\partial^{2}\Lambda}{\partial b\partial T}&-\frac{\partial^{2}\Lambda}{ \partial T^{2}}\end{bmatrix} \] \[=\frac{1}{\frac{\partial^{2}\Lambda}{\frac{\partial^{2}\Lambda}{ \partial T^{2}}\frac{\partial^{2}\Lambda}{\partial b^{2}}-\left(\frac{ \partial^{2}\Lambda}{\partial T\partial b}\right)^{2}}} \begin{bmatrix}-\frac{ \partial^{2}\Lambda}{\partial b^{2}}&\frac{\partial^{2}\Lambda}{\partial T \partial b}\\ \frac{\partial^{2}\Lambda}{\partial b\partial T}&-\frac{\partial^{2}\Lambda}{ \partial T^{2}}\end{bmatrix}  \tag{29}\]  
with  
\[\frac{\partial^{2}\Lambda}{\partial T^{2}} =\frac{b}{T^{2}}\Bigg{(}n-m-(b+1)\sum_{i=1}^{n}\left(\frac{t_{i}} {T}\right)^{b}\Bigg{)} \tag{30}\] \[\frac{\partial^{2}\Lambda}{\partial b^{2}} =\frac{m-n}{b^{2}}-\sum_{i=1}^{n-m}\left(\ln\left(\frac{t_{i}}{T} \right)\right)^{2}\cdot\left(\frac{t_{i}}{T}\right)^{b}+\sum_{j=1}^{m}\left( \ln\left(\frac{t_{j}}{T}\right)\right)^{2}\cdot\left(\frac{t_{j}}{T}\right)^{b}\] (31) \[\frac{\partial^{2}\Lambda}{\partial T\partial b} =\frac{\partial^{2}\Lambda}{\partial b\partial T}=\frac{m-n}{T}+ \frac{1}{T}\sum_{i=1}^{n}\left(1+b\ln\left(\frac{t_{i}}{T}\right)\right)\cdot \left(\frac{t_{j}}{T}\right)^{b} \tag{32}\]  
and \(\text{Var}(\cdot)\) and \(\text{Cov}(\cdot,\cdot)\) being the variance and covariance, respectively. The determinant of a matrix is \(\det(\cdot)\). The summation in (30) and (32) is over both the uncensored failure times \(t_{i}\) and the censored failure times \(t_{j}\). Only a single summation index \(i\) is used. Using a linear Taylor series expansion [8; 59], the variance of the MLE of the quantile function for the Weibull distribution \(t_{q}=T\left(-\ln(1-q)\right)^{1/b}\) can be approximated by  
\[\text{Var}(t_{q}) =\bigg{(}\frac{\partial t_{q}}{\partial T}\bigg{)}^{2}\text{Var}(T )+\bigg{(}\frac{\partial t_{q}}{\partial b}\bigg{)}^{2}\text{Var}(b)\ \ +2\,\frac{\partial t_{q}}{\partial T}\,\frac{\partial t_{q}}{ \partial b}\,\text{Cov}(T,b)\] \[=(-\ln(1-q))^{2/b}\,\text{Var}(T)\ \ +\frac{T^{2}}{b^{4}}\ln(-\ln(1-q))^{2}(-\ln(1-q))^{2/b}\, \text{Var}(b)\] \[\ \ -\frac{2T}{b^{2}}\ln(-\ln(1-q))(-\ln(1-q))^{2/b}\,\text{Cov}(T,b). \tag{33}\]  
According to the asymptotic behavior of the MLE and the CLT under certain regularity conditions [8], the variable \(t_{q}\) approximately follows a normal distribution; hence,  
\[t_{q}\sim\mathcal{N}\Big{(}T(-\ln(1-q))^{1/b},\,\sqrt{\text{Var}(t_{q})}\Big{)}. \tag{34}\]  
As no bootstrap and no MCS take place, the failure times in (30)-(32) can be calculated using the Weibull failure distribution \(F(t)\) of the prior knowledge. Using order statistics [54], the median of each failure time as a function of the sample size is  
\[t_{i}=F^{-1}\Big{(}F^{-1}_{\text{beta}}\left(0.5;\,i,n-i+1\right)\Big{)} \tag{35}\]  
with \(F_{\text{beta}}(q;\,A,B)\) being the quantile function (inverse) of the beta distribution for quantile \(q\) and parameters \(A\) and \(B\).