## 7 Outlier Identification and Exclusion  
The RAIM is a commonly used technique in the GNSS receivers for assurance of integrity and reliability of the estimated position. The fundamental task of the RAIM is the identification and exclusion of outlying observations for error free positioning. In the classic RAIM approach, the least-squares residual is used as the test statistic for outlier identification. As a rule of thumb, if the sum of squared residuals follows the chi-squared distribution, presence of outlier is rejected, otherwise observations are verified for outlier and the observation with the highest residual error is considered as the outlier. The primary drawback of this approach is that it is also possible that a wrong estimated position or exclusion of a wrong observation may lead to successful chi-square test. Conversely, in this paper, our proposed estimation technique itself maintains the integrity of estimated position by selection of the best fit observations. If \(\overline{\epsilon}\) proportion of the observations are outliers, the probability of the outlier free observation is \(\overline{\mu}=\left(1-\overline{\epsilon}\right)^{n_{sub}}\), \(n_{sub}\) is the number of observations in one subsample and the probability of at least one good subsample is \(1-\left(1-\overline{\mu}\right)^{\overline{N}}\), \(\overline{N}\) is the total number of subsamples. However, to enhance the positioning accuracy and reliability, we also proposed a two-step outlier detection and exclusion process. The proposed outlier detection and exclusion scheme is explained in the following,  
1. A predetection test is performed to predict the possible outlier observations prior to the FDE (fault detection and exclusion) process. A predefined threshold level is used to predict the outlier observations based on the confidence level and on the confidence interval. A confidence interval is the range of values those act as the good estimates and the confidence level is the proportion of the confidence interval that contains the true value of their corresponding parameter. The confidence interval can be calculated from data [35; 36]. However, the desired confidence level is defined by the user. Let \(cl\) is the confidence level in predetection test, a threshold \(th_{cl}\) can be computed using normal inverse cumulative distribution function (also called quantile function). \[t_{cl}=F^{-1}\left(\kappa\middle|\widetilde{\mu},\widetilde{\sigma}\right)= \left\{t_{cl}:F\left(t_{cl}\middle|\widetilde{\mu},\widetilde{\sigma}\right)= \kappa\right\}\] (31)Here \(\kappa=(cl+1)/2\) is the probability for corresponding confidence level with mean \(\widetilde{\mu}\) and standard deviation \(\widetilde{\sigma}\) and \(F\big{(}t_{cl}|\widetilde{\mu},\widetilde{\sigma}\big{)}=\frac{1}{\widetilde{ \sigma}\sqrt{2\pi}}\int\limits_{-\infty}^{t_{cl}}e^{\frac{-(t-\widetilde{\
u} )^{2}}{2\widetilde{\sigma}^{2}}}dt\) is the cumulative distribution function (CDF). The scaled observation residual is computed as, \[\overline{u}_{sc}=\overline{S}Y\] (32) Here \(Y=(\widetilde{\rho}-\hat{\rho})/s_{sc}\) and \(\overline{S}=I-\mathbf{H}\big{(}\mathbf{H}^{\mathrm{T}}\mathbf{H}\big{)}^{- 1}\mathbf{H}^{\mathrm{T}}\). As soon as \(\gamma=\sum\limits_{l=1}^{n+m}\overline{u}_{sc,l}^{2}\) follows \(\chi^{2}\) distribution with \(\overline{\mathrm{n}}-\overline{v}\) degree of freedom, the estimated position is considered as reliable, where is the number of observations used for integrity monitoring and \(\overline{v}\) is the number of unknown to be estimated. 2. Otherwise, a set of observations is selected as the potential outliers based on the following relation, \[\big{\{}\rho_{l}\in\{outliers\}\big{|}\overline{u}_{sc,l}>t_{cl}\forall \overline{u}_{sc}\wedge\rho_{l}\leftrightarrow\overline{u}_{sc}\big{\}}\] (33) Let \(\varphi=\underset{\overline{u}_{sc,l}>t_{cl}}{\arg}\big{(}\rho_{outlier,k_{ 1}},\rho_{outlier,k_{2}},\ldots,\rho_{outlier,k_{n}}\big{)}\) is the set of predicted outlier observations. \(k_{l}\) is corresponding observation number.