## 2 Conventional Reliability Theory  
The null hypothesis \(\mathcal{H}_{0}\) corresponds to a model in the absence of outliers. Thus, the null hypothesis of the standard Gauss-Markov model in linear or linearised form is given by [24]:  
\[\mathcal{H}_{0}:\mathbb{E}\{\mathbf{y}\}=\mathbf{A}\mathbf{x}+\mathbb{E}\{\mathbf{e}\}=\mathbf{A} \mathbf{x}, \tag{1}\]  
where \(\mathbb{E}\{.\}\) is the expectation operator, \(\mathbf{y}\in~{}\mathbb{R}^{n}\) the vector of measurements, \(\mathbf{A}\in~{}\mathbb{R}^{n\times u}\) the _Design Matrix_) (also referred as Jacobian matrix) of full rank \(u\), \(\mathbf{x}\in~{}\mathbb{R}^{u}\) the unknown parameter vector, and \(\mathbf{e}\in~{}\mathbb{R}^{u}\) the unknown vector of measurement errors. Typically, it is assumed that the errors of the good measurements are normally distributed with expectation zero [32], i.e.,:  
\[\mathbf{e}\sim\mathbf{N}(\mathbf{0},\mathbf{Q_{e}}), \tag{2}\]  
with a known positive definite symmetric covariance matrix \(\mathbf{Q_{e}}\in~{}\mathbb{R}^{n\times n}\). Here, we confine ourselves to the case that \(\mathbf{A}\) and \(\mathbf{Q_{e}}\) have full column rank. The redundancy (or degrees of freedom) of the model in Equation (1) is \(r=n-u\). However, any model is only an approximation to the truth. This implies that we inevitably encounter misspecified models. In contrast to the \(\mathcal{H}_{0}\), [PERSON] [4] introduced a mean shift model that defines the alternative hypothesis \(\mathcal{H}_{A}\), also referred to as model misspecification, as follows:  
\[\mathcal{H}_{A}:\mathbb{E}\{\mathbf{y}\}=\mathbf{A}\mathbf{x}+\mathbf{c_{i}}\
abla_{i}+ \mathbb{E}\{\mathbf{e}\}= \begin{pmatrix}\mathbf{A}&\mathbf{c_{i}}\end{pmatrix}\begin{pmatrix} \mathbf{x}\\ \
abla_{i}\end{pmatrix} ,\forall i=1,\cdots,n \tag{3}\]  
where \(\mathbf{c_{i}}\) takes the form of a canonical unit vector with \(1\) as its \(i\)th entry and zeros elsewhere. The \(1\) value means that an \(i\)th bias parameter of magnitude \(\
abla_{i}\) affects an \(i\)th measurement and \(0\) otherwise. We have,for instance, \(\mathbf{c_{i}}\)=\( \begin{pmatrix}0&0&0&\cdots&1^{i_{th}}&0&\cdots&0\end{pmatrix} ^{T}\). In other words, \(\mathbf{c_{i}}\) specifies the type of model error and \(\
abla_{i}\) the size of the model error, or outlier. The likelihood ratio test to test \(\mathcal{H}_{0}\) against \(\mathcal{H}_{A}\) is given by:  
\[\text{\emph{Accept}}\ \ \mathcal{H}_{0}\ \ if\ \ |\mathbf{w_{i}}|\leq k=\Phi^{-1}(1- \frac{\alpha}{2}),\ \text{\emph{reject}}\ \text{\emph{otherwise}}\ \ \text{\emph{in}}\ \ \text{\emph{ favour}}\ \text{\emph{of}}\ \ \mathcal{H}_{A} \tag{4}\]  
and the test statistics (known as [PERSON]'s \(w\)-test) are the normalised least-squares residuals given by [4]:  
\[\mathbf{w_{i}}=\frac{\mathbf{c_{i}}^{T}\mathbf{Q_{e}^{-1}\hat{\mathbf{e}}}}{\sqrt{\mathbf{c_{i}}^{ T}\mathbf{Q_{e}^{-1}\mathbf{Q_{e}}\mathbf{Q_{e}^{-1}\mathbf{c_{i}}}}}} \tag{5}\]  
According to (4) and (5), we have:  
* \(k\) is the critical value. The critical value \(k\) is the the tabular value from the cumulative distribution function (cdf) of the standard normal \(\mathbf{N}(0,1)\) based on the chosen of a significance level \(\alpha\). Because we perform a two-sided test of the form \(|\mathbf{w_{i}}|\leq k\) we have \(\alpha/2\). For example, for \(\alpha=0.01\), we obtain \(k=2.576\). In this case, if \(|\mathbf{w_{i}}|>2.576\) for some \(y_{i}\) one may reject \(\mathcal{H}_{0}\). * \(\Phi^{-1}\) denotes the inverse of the normal cumulative distribution function. * \(\hat{\mathbf{e}}\) is the least-squares residuals vector under \(\mathcal{H}_{0}\) and \(\mathbf{Q_{\hat{e}}}\) the covariance matrix of the best linear unbiased estimator of \(\hat{\mathbf{e}}\) under \(\mathcal{H}_{0}\). The decision rule in (4) says that if the test statistic \(|w_{i}|\) in (5) is larger than some critical value \(k\), i.e., a percentile of its probability distribution, then we reject the null hypothesis \(\mathcal{H}_{0}\) in favour of the alternative hypothesis \(\mathcal{H}_{A}\).