Methods for reducing the resources needed to detect and identify faulty pseudorange measurements in a GNSS receiver are described. In a Receiver Autonomous Integrity Monitoring (RAIM) method, a position solution is calculated using a weighted least squares method on measurements from satellites of one or more GNSS constellations. A test statistic is calculated from residuals and a threshold is calculated based on a probability function. If the test statistic is greater than or equal to the threshold, a subset is selected from the set of pseudorange measurements using a metric indicative of possible measurement error. A measurement is selected from the subset using a metric indicative of signal strength or some other metric and discarded from the set of pseudorange measurements. If the number of measurements remaining in the set of pseudorange measurements is greater than five, the method loops back to the step of calculating a position solution.