# 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 general method for calculating \(P_{\text{ts}}\) using the bootstrap approach can become time- and resource-consuming if several different tests need to be analyzed and high accuracy is to be assured by a high iteration number. In order to enable a fast way of calculating \(P_{\text{ts}}\) for failure-based tests (EoL test) in a way that is effortless to implement, an approximate calculation can be sufficient. To obtain the distributions of \(\tau_{H_{0}}\) and \(\tau_{H_{1}}\) without an MCS or bootstrap, the central limit theorem (CLT) [56; 57] can be used. Several statistical phenomena are normally distributed if the sample size is very large (\(n\rightarrow\infty\)). If the sample size is finite, the normal distribution can nonetheless be used as a rough approximation. Using the CLT, the distribution of the sample quantile of a known distribution \(F(t)\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), as follows [57; 58]:  
\[\mu =F^{-1}(q) \tag{22}\] \[\sigma =\sqrt{\frac{q\cdot(1-q)}{n\cdot f\left(F^{-1}(q)\right)^{2}}} \tag{23}\]  
where \(q\) is the proportion of the quantile, \(F^{-1}(q)\) is the inverse of \(F(t)\), \(f(t)\) is the pdf of \(F(t)\), and \(n\) the sample size. To abbreviate (22) and (23), a normally distributed variable \(t\) can be written as \(t\sim\mathcal{N}(\mu,\sigma)\). Using the test statistic from (5) and the asymptotic behavior from (22) and (23), the approximated distributions \(\tau_{H_{0}}\) and \(\tau_{H_{1}}\) are the following normal distributions:  
\[\tau_{H_{0}} \sim\mathcal{N}\left(0,\,\sqrt{\frac{R_{\text{r}}\cdot(1-R_{\text {r}})}{n\cdot f_{0}(t_{\text{r}})^{2}}}\right) \tag{24}\] \[\tau_{H_{1}} \sim\mathcal{N}\left(t_{\text{p}}-t_{\text{r}},\,\sqrt{\frac{R_{ \text{r}}\cdot(1-R_{\text{r}})}{n\cdot f\left(t_{\text{p}}\right)^{2}}}\right) \tag{25}\]  
with \(f_{0}(t)\) as the shifted pdf of the failure distribution of prior knowledge \(f(t)\), such that it is valid under \(H_{0}\) and \(t_{\text{p}}=F^{-1}(1-R_{r})\). Where a Weibull distribution is concerned as the failure distribution of prior knowledge \(F(t)\) with scale parameter \(T\) and shape parameter \(b\), the scale parameter of \(f_{0}(t)\) can be calculated as  
\[T_{0}=(1-s)\,T=\frac{t_{\text{r}}}{t_{\text{p}}}\,T \tag{26}\]while the shape parameter stays the same (\(b_{0}=b\)). Using the CLT, (24) and (25), the Probability of Test Success \(P_{\text{ts}}\) of an EoL test, if the lifetime quantile is determined by the empiric sample quantile, can be calculated approximately by  
\[P_{\text{ts}}\approx 1-\Phi\Bigg{(}\Phi^{-1}\Bigg{(}C_{\text{r}};0,\sqrt{ \frac{R_{\text{r}}\cdot(1-R_{\text{r}})}{n\cdot f_{0}(t_{\text{r}})^{2}}} \Bigg{)},\ t_{\text{p}}-t_{\text{r}},\ \sqrt{\frac{R_{\text{r}}\cdot(1-R_{\text{r}})}{n \cdot f(t_{\text{p}})^{2}}}\Bigg{)} \tag{27}\]  
with \(\Phi(x;\mu,\sigma)\) being the cumulative distribution function (cdf) of the normal distribution at the value \(x\) with the parameters \(\mu\) and \(\sigma\). The inverse of the cdf of the normal distribution is \(\Phi^{-1}(q;\mu,\sigma)\). The approximation of \(P_{\text{ts}}\) uses the quantile of the sample instead of the quantile one would obtain using a distribution estimation. If a Weibull distribution and an MLE fit are concerned, a different approximation can be derived using the CLT and the likelihood, as well as the variance-covariance matrix together with a Taylor series expansion. Therefore, the \(P_{\text{ts}}\) of the most common censored or uncensored EoL tests for Weibull distributions using an MLE estimation can be approximately calculated without the need for an MCS or bootstrap approach. The log-likelihood \(\Lambda\) of the two-parameter Weibull distribution is [8]  
\[\Lambda(T,b)=\ (n-m)(\ln(b)-b\ln(T))+\sum_{i=1}^{n-m}\Bigg{(}(b-1)\ln(t_{i})- \left(\frac{t_{i}}{T}\right)^{b}\Bigg{)}-\sum_{j=1}^{m}\left(\frac{t_{j}}{T} \right)^{b} \tag{28}\]  
with \(n-m\) uncensored failure times \(t_{i}\) and \(m\) right-censored failure times \(t_{j}\).