Future multi-constellation Global navigation satellite system (GNSS) will provide a large number of redundant ranging signals, which will improve the performance of receiver autonomous integrity monitoring (RAIM), but it will also increase the probability of satellite faults, thereby increasing the continuity risk. In response, in this paper, a new Chi-Squared (Chi2) RAIM approach to fault detection and exclusion (FDE) is developed, and compared to Solution Separation (SS) RAIM. The paper first introduces complete integrity and continuity risk equations for Chi2 RAIM FDE, which are currently missing in the literature. Probability bounds are developed, which express the fact that the reduction in continuity risk obtained using exclusion comes at the cost of a higher integrity risk. It is then shown that the Chi2 approach can provide a tighter integrity risk bound as compared to SS, but the SS bound enables risk evaluation in practical implementations where computation resources are limited. Finally, the SS and Chi2 FDE integrity and continuity risk bounds are implemented to establish worldwide availability maps for Advanced RAIM (ARAIM) in an example aircraft approach application using GPS, Galileo, GLONASS and Beidou satellite constellations.