This article develops statistics based on the Kullback–Leibler (KL) divergence to monitor large-scale technical systems. These statistics detect anomalous system behavior by comparing estimated density functions for the current process behavior with reference density functions. For Gaussian distributed process variables, the paper proves that the difference in density functions, measured by the KL divergence, is a more sensitive measure than existing work involving multivariate statistics. To cater for a wide range of potential application areas, the paper develops monitoring concepts for linear static systems, that can produce Gaussian as well as non-Gaussian distributed process variables. Using recorded data from a glass melter, the article demonstrates the increased sensitivity of the KL-based statistics by comparing them to competitive ones.