## 2 Conventional Reliability Theory
Because \(w\)-test is based on binary hypothesis testing, in which one decides between the null hypothesis \(\mathcal{H}_{0}\) and a single alternative hypothesis \(\mathcal{H}_{A}\), it may lead to type I decision error (\(\alpha\)) and type II decision error (\(\beta\)). The probability level \(\alpha\) is the probability of rejecting the null hypothesis when it is true, whereas \(\beta\) is the probability of failing to reject the null hypothesis when it is false. The complements of \(\alpha\) and \(\beta\) are well-known confidence level (\(CL\)) and the statistical power (\(\gamma\)), respectively. The first deals with the probability of accepting a true null hypothesis; the second, with the probability of correctly accepting the alternative hypothesis. In other words, we have \(CL=1-\alpha\) and \(\gamma=1-\beta\). The normalised least-squares residual \(w_{i}\) follows a standard normal distribution with the expectation that \(\mu=0\) if \(\mathcal{H}_{0}\) holds true. On the other hand, if the system is contaminated with a single error at the \(i\)th position of the dataset, there is an outlier that causes the expectation of \(w_{i}\) to become \(\mu>0\). The effect can be best understood using the non-central chi-squared distribution with one degree of freedom (i.e., for a single outlier). Under the alternative hypothesis \(\mathcal{H}_{A}\), the expectation of \(w_{i}\) is the square-root of the non-centrality parameter \(\lambda_{q=1}\) from the chi-square distribution with one degree of freedom (\(q=1\)), which is given by:  
\[\mathbb{E}\{w_{i}\}=\sqrt{\lambda_{q=1}}=\sqrt{\mathbf{c_{i}}^{T}\mathbf{Q_{e}^{-1} \mathbf{Q_{\hat{e}}}\mathbf{Q_{e}^{-1}\mathbf{c_{i}}\
abla_{i}^{2}}}} \tag{6}\]  
where \(\lambda_{q=1}\) is the non-centrality parameter for one degree of freedom \(q=1\). Under \(\mathcal{H}_{A}\), the expectation of \(w_{i}\) is \(\sqrt{\lambda_{q=1}}\), which is caused by the presence of an outlier. The non-centrality parameter \(\lambda_{q=1}\) in Equation (6) describes the expectation of a \(w\)-test when the null hypothesis \(\mathcal{H}_{0}\) is false (so the alternative hypothesis \(\mathcal{H}_{A}\) is true). This leads to their use in calculating statistical power. The term \(\mathbf{c_{i}}^{T}\mathbf{Q_{e}^{-1}\mathbf{Q_{\hat{e}}}\mathbf{Q_{e}^{-1}\mathbf{c_{i}}}}\) in Equation (6) is a scalar for \(q=1\) and therefore it can be rewritten as follows [30]:  
\[|\
abla_{i}|=MDB_{(i)}=\sqrt{\frac{\lambda_{q=1}}{\mathbf{c_{i}}^{T}\mathbf{Q_{e}^{-1} \mathbf{Q_{\hat{e}}}\mathbf{Q_{e}^{-1}\mathbf{c_{i}}}}}},\ \forall i=1,\cdots,n \tag{7}\]where \(|\
abla_{i}|\) is the Minimal Detectable Bias \(MDB_{(i)}\), which can be computed for each measurement of the \(n\) alternative hypotheses according to Equation (3). For a single outlier, the variance of estimated outlier, denoted by \(\sigma^{2}_{\
abla_{i}}\), is:  
\[\sigma^{2}_{\
abla_{i}}=\left(\mathbf{c_{i}}^{T}\mathbf{Q_{e}}^{-1}\mathbf{Q_{e}}\mathbf{Q_{e}} ^{-1}\mathbf{c_{i}}\right)^{-1},\ \forall i=1,\cdots,n \tag{8}\]  
Thus, the MDB can also be written as:  
\[MDB_{(i)}=\sigma_{\
abla_{i}}\sqrt{\lambda_{q=1}},\ \forall i=1,\cdots,n \tag{9}\]  
where \(\sigma_{\
abla_{i}}=\sqrt{\sigma^{2}_{\
abla_{i}}}\) is the standard-deviation of estimated outlier \(\
abla_{i}\). The MDB in Equations (7)-(9) of an alternative hypothesis is the smallest magnitude outlier that can lead to rejection of a null hypothesis for a given \(\alpha\) and \(\beta\). Thus, for each model of the alternative hypothesis \(\mathcal{H}_{A}\), the corresponding MDB can be computed. The key point of MDB is that it can work as a tool for designing systems capable of withstanding outlier with a certain degree of probability. The non-centrality parameter \(\lambda_{q=1}\) can be computed as a function of type 1 decision error \(\alpha\), type 2 decision error \(\beta\) and the degrees of freedom of the test \(q\). To obtain the non-centrality parameter \(\lambda_{q=1}\), here we use the recursive algorithm based on the work by [33], namely bisection algorithm.