# 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 approximation of the median of the beta distribution of Benard [60], commonly used in rank regression [10], can be used here for a simpler implementation; hence, for a Weibull distribution with parameters \(T\) and \(b\) from prior knowledge,  
\[t_{i}\approx T\cdot\bigg{(}-\ln\bigg{(}1-\frac{i-0.3}{n+0.4}\bigg{)}\bigg{)}^{ 1/b}\ \ \ \ \forall i\in[1,n]. \tag{36}\]  
Censored failure times can be generated using (35) or (36) and the corresponding censoring scheme. The established distribution of the sample quantile using the Weibull distribution of prior knowledge in (34) corresponds to the alternative hypothesis \(H_{1}\) of (2). Therefore, the approximated distribution of the quantile \(t_{R_{\text{r}},H_{1}}\) under the validity of the alternative hypothesis is  
\[t_{R_{\text{r}},H_{1}}\sim\mathcal{N}\Big{(}T(-\ln(R_{\text{r}}))^{1/b},\, \sigma_{H_{1}}\Big{)} \tag{37}\]  
with  
\[\frac{\partial^{2}\Lambda}{\partial b^{2}} =\frac{m-n}{b^{2}}-\sum_{i=1}^{n-m}\left(\ln\left(\frac{t_{i}}{T} \right)\right)^{2}\cdot\left(\frac{t_{i}}{T}\right)^{b}+\sum_{j=1}^{m}\left( \ln\left(\frac{t_{j}}{T}\right)\right)^{2}\cdot\left(\frac{t_{j}}{T}\right)^{b}\]0  
using (29)-(32) with the parameters of the Weibull distribution from the prior knowledge as well as the failure times of (35) or (36) and \(q=1-R_{\text{r}}\) in (33). The distribution of the quantile \(t_{R_{\text{r}},H_{0}}\) under the validity of the null hypothesis has to use the shifted failure distribution with scale parameter \(T_{0}\) of (26) and the calculated failure times using this shifted failure distribution and (35) or (36); accordingly,  
\[\frac{\partial^{2}\Lambda}{\partial b^{2}} =\frac{m-n}{b^{2}}-\sum_{i=1}^{n-m}\left(\ln\left(\frac{t_{i}}{T} \right)\right)^{2}\cdot\left(\frac{t_{i}}{T}\right)^{b}+\sum_{j=1}^{m}\left( \ln\left(\frac{t_{j}}{T}\right)\right)^{2}\cdot\left(\frac{t_{j}}{T}\right)^{b}\]1with  
\[\frac{\partial^{2}\Lambda}{\partial b^{2}} =\frac{m-n}{b^{2}}-\sum_{i=1}^{n-m}\left(\ln\left(\frac{t_{i}}{T} \right)\right)^{2}\cdot\left(\frac{t_{i}}{T}\right)^{b}+\sum_{j=1}^{m}\left( \ln\left(\frac{t_{j}}{T}\right)\right)^{2}\cdot\left(\frac{t_{j}}{T}\right)^{b}\]2  
where \(\text{Var}(T_{0},b)\) and \(\text{Cov}(T_{0},b)\) do not correspond to a different variable. Instead, (29) is used at the corresponding value of \(T_{0}\) for \(T\). Using these equations (from (37) to (40)), the \(P_{\text{ts}}\) of a right-censored or uncensored EoL test in which the lifetime quantile is determined by distribution estimation via MLE can be calculated approximately by  
\[\frac{\partial^{2}\Lambda}{\partial b^{2}} =\frac{m-n}{b^{2}}-\sum_{i=1}^{n-m}\left(\ln\left(\frac{t_{i}}{T} \right)\right)^{2}\cdot\left(\frac{t_{i}}{T}\right)^{b}+\sum_{j=1}^{m}\left( \ln\left(\frac{t_{j}}{T}\right)\right)^{2}\cdot\left(\frac{t_{j}}{T}\right)^{b}\]3  
The approximations of (27) and (41) are approximate in several ways: first, (27) and (41) approximate the desired lifetime quantile distribution via a normal distribution, which does not hold for small sample sizes; second, the failure times used in (41) are approximated using order statistics in addition to the Benard approximation to the median of the beta distribution. However, the approximate calculations can be implemented effortlessly, as even most spreadsheet programs of the popular operating systems of personal computers have implementations of the normal distribution. The approximations are of benefit in cases where several test scenarios are required or dependencies over a large range of values need to be found. If the feasible space of test scenarios is narrowed down using the approximate calculation, the more accurate general method can be used to identify the optimal test.