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Besides accuracy, integrity is another important performance measure of GNSS. The classical least-squares-residual (LSR) method and parity vector (PV) method are often used in the receiver autonomous integrity monitoring (RAIM). The two fault detection methods assume that the observation errors of different satellites are the same, ignoring possible variations of accuracy between observations. In this study, the mathematical models of the weighted least-squares-residual (WLSR) method and the weighted parity vector (WPV) method are derived in detail. The equivalence of the two methods is established with statistical tests. The WPV method is applied to detect those faults based on both GPS and BDS observations collected at Wuhan JiuFeng Station (JFNG). The theoretical results show that this method has lower computational complexity than the WLSR method, hence more suited for cases requiring fast fault detection. The fault detection rate increases as the deviation of the pseudorange observation increases. Thus, using the threshold value <inline-formula> <tex-math notation="LaTeX">$T_{d} $ </tex-math></inline-formula> of the posterior unit weight error <inline-formula> <tex-math notation="LaTeX">$\hat {\sigma }_{0} $ </tex-math></inline-formula>, the WPV achieves a higher fault detection rate than using a priori unit weight error <inline-formula> <tex-math notation="LaTeX">$\sigma _{0} $ </tex-math></inline-formula>. The experiments show that these two methods can detect relatively large faults, it is possible to detect them in GPS observations if <inline-formula> <tex-math notation="LaTeX">$\sigma _{0} $ </tex-math></inline-formula> is more than <inline-formula> <tex-math notation="LaTeX">$12\times $ </tex-math></inline-formula>bias (<inline-formula> <tex-math notation="LaTeX">$1\times $ </tex-math></inline-formula>bias=8 m) and <inline-formula> <tex-math notation="LaTeX">$\hat {\sigma }_{0} $ </tex-math></inline-formula> superior to <inline-formula> <tex-math notation="LaTeX">$4\times $ </tex-math></inline-formula>bias, whereas the faults detection in BDS observations requires a deviation bigger than <inline-formula> <tex-math notation="LaTeX">$8\times $ </tex-math></inline-formula>bias and <inline-formula> <tex-math notation="LaTeX">$6\times $ </tex-math></inline-formula>bias, respectively. But these two methods are insensitive when the deviation is smaller.

Equivalence Proof and Performance Analysis of Weighted Least Squares Residual Method and Weighted Parity Vector Method in RAIM