Future multi-constellation GNSS open the possibility to fulfill stringent navigation integrity requirements specified in safety-critical applications using receiver autonomous integrity monitoring (RAIM). In this paper, both the RAIM detector and its estimator are analyzed to develop a new algorithm. In the first part of this paper, the detector is selected by rigorously comparing two of the most widely implemented methods. In particular, the paper reveals fundamental differences between solution separation (SS) and residual-based (RB) RAIM. SS provides higher fault-detection performance than RB RAIM because the SS test statistic is tailored to the fault hypothesis, and to the state of interest. To prove these results in presence of multi-measurement faults, which occur in multi-constellation GNSS, analytical expressions of the worst-case fault direction are derived for both SS and RB RAIM. In the second part of the paper, a non-least-squares estimator is designed to reduce the integrity risk at the cost of lower accuracy performance for applications where integrity requirements are more demanding than accuracy requirements. The new estimator is numerically determined by solving an integrity risk minimization problem that includes multiple simultaneous fault hypotheses. Performance analyses show a substantial drop in integrity risk using the new RAIM algorithm as compared to a SS method that uses a least-squares estimator. In parallel, the decrease in accuracy performance is quantified. Combined availability of accuracy and integrity is evaluated at an example location for a GPS/Galileo navigation system.