This paper introduces a new Kalman filter-based method for detecting sensor faults in linear dynamic systems. In contrast with existing sequential fault-detection algorithms, the proposed method enables direct evaluation of the integrity risk, which is the probability that an undetected fault causes state estimate errors to exceed predefined bounds of acceptability. The new method is also computationally efficient and straightforward to implement. The algorithm's detection test statistic is established in three steps. First, the weighted norms of current and past-time Kalman filter residuals are defined as generalized non-centrally chi-square distributed random variables. Second, these residuals are proved to be stochastically independent from the state estimate error. Third, current-time and past-time residuals are shown to be mutually independent, so that the Kalman filter-based test statistic can be recursively updated in real time by simply adding the current-time residual contribution to a previously computed weighted norm of past-time residuals. The Kalman filter-based integrity monitor is evaluated against worst-case fault profiles, which are also derived in this paper. Finally, performance analyses results are presented for an example application of aircraft precision approach navigation, where differential ranging signals from a multi-constellation satellite navigation system are filtered for positioning and carrier phase cycle ambiguity estimation.