Optimality of the mean-square value, skewness, kurtosis, and synentropy for detection of weak additive fault signatures in (nearly) Gaussian noise

The mean-square value, the skewness, and the kurtosis are scalar indicators widely used in health monitoring systems. They are simple to use and have proven sensitive to fault occurrence. This paper theoretically proves that they are the optimal health indicators for the early detection of faults in additive Gaussian noise, in the sense of maximising the probability of detection given a constant false alarm rate. The mean-square value is optimal in scenarios where the scale parameter of the background noise is known; otherwise, it should be replaced by a new indicator, coined “synentropy”. Remarkably, these results hold true independently of the fault signature. The paper also shows how to correct these indicators when the background noise slightly departs from Gaussianity, provides their statistical thresholds, and evidences that their relative sensitivity depends on the data length. • The MS value, the skewness and the kurtosis are widely used in condition monitoring. • They maximise the probability of detection of a weak fault in additive Gaussian noise. • They are related to the Hermite moments. • The synentropy is introduced as a new entropic measure. • The MS value, skewness, kurtosis and synentropy define optimal health indicators.