## 2 Integrity Risk Estimation for RAIM  
Integrity risk, also called the probability of hazardous misleading information (PHMI), is defined as the probability that a position error exceeds the alarm limit (AL) or the protect levels and test statistic remain below the detection threshold [16]:  
\[IR=P(|\delta|>l\cap t<T) \tag{1}\]  
where \(\delta\) represents the position error; \(l\) is the alarm limit; \(t\) and \(T\) are the test statistic and detection threshold of RAIM, respectively. The integrity risk of RAIM based on (1) can be estimated when given the statistical distribution of position error \(\delta\) and test statistic \(t\). Generally, the nominal observation error is assumed to have a zero mean Gaussian distribution with a STD observation error \(\sigma\) determined by \(URA\), and the troposphericdelay and the user-related errors. With the least-squares estimator, the position error and test statistic distributions can be characterized by  
\[\delta_{q,i}\sim N\Big{(}b_{i}\Big{|}\mathbf{A}_{q,i}\Big{|},\sigma_{q}^{2}\Big{)},t_ {i}\sim\chi^{2}(\lambda_{i},n-m) \tag{2}\]  
where the subscript \(i\) and \(q\) indicate the \(i\)th satellite and the \(q\)th state, respectively; \(n\) is the number of satellite signals tracked; \(m\) is the number of states to be estimated; \(b_{i}\) is the fault-induced bias of the \(i\)th satellite and is absent under the nominal mode, where the position error and test statistic follow the zero mean Gaussian distribution and chi-square distribution, respectively. Under the fault mode, the position error follows the non-zero mean Gaussian distribution, and the test statistic follows the non-central chi-square distribution. \(\mathbf{A}=\Big{(}\mathbf{G}^{T}\mathbf{G}\Big{)}^{-1}\mathbf{G}^{T}\) is the transformation matrix from the observation domain to the position domain; \(\mathbf{G}\) is the geometry matrix determined by satellite geometry and the user's location; \(\sigma_{q}=\sigma\sqrt{\Big{(}\mathbf{G}^{T}\mathbf{G}\Big{)}_{q,i}^{-1}}\) is the STD observation error for state \(q\); \(\lambda_{i}=(b_{i}/\sigma)^{2}\mathbf{S}_{i,i}\) is the non-centrality parameter; and \(\mathbf{S}=\mathbf{I}-\mathbf{G}\Big{(}\mathbf{G}^{T}\mathbf{G}\Big{)}^{-1}\mathbf{G}^{T}\) is the transformation matrix from the observation domain to the RAIM detection domain. The statistical distribution of position error and test statistic is independent of each other based on the least-squares estimator [17]. The integrity risk can be computed under the mutually exclusive and exhaustive fault mode  
\[IR=\sum\limits_{i=0}^{n}P(|\delta_{i}|>l|\mathrm{H}_{i})P(t_{i}<T|H_{i})P(H_{i}) \tag{3}\]  
where \(P(H_{i})\) is the probability of the fault mode \(H_{i}\), which may be the nominal or fault mode. The integrity risk can be computed as the product of the probability of position error exceeding the alarm limit and the probability of the test statistic beyond the detection threshold. Provided the probability of the \(i\)th satellite fault is \(P_{\text{sat},i}\), the integrity risk in (3) can be expressed as follows under the single-satellite fault mode:  
\[ \begin{array}{ll}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{ rgb}{0,0}\pgfsys@color@gray@stroke{0}{0}\pgfsys@color@gray@fill{0}{0}{I}R}=&{ \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \pgfsys@color@gray@stroke{0}{0}\pgfsys@color@gray@fill{0}{0}{\sum\limits_{i=1}^{n }}}\big{[}Q\big{(}z_{i}^{+}\big{)}+Q\big{(}z_{i}^{-}\big{)}\big{]}P(T,n-m, \lambda_{i})P_{\text{sat},i}\prod\limits_{k=1,k\
eq i}^{n}(1-P_{\text{sat}, k})\\ &+2Q(z_{0})P(T,n-m)\prod\limits_{k=1}^{n}\big{(}1-P_{\text{sat},k})\end{array}