Anomaly detection/detectability for a linear model with a bounded nuisance parameter

Anomaly detection is addressed within a statistical framework. Often the statistical model is composed of two types of parameters: the informative parameters and the nuisance ones. The nuisance parameters are of no interest for detection but they are necessary to complete the model. In the case of unknown, non-random and non-bounded nuisance parameters, their elimination is unavoidable. Some approaches based on the assumption that the nuisance parameters belonging to a subspace interfere with the informative ones in a linear manner, use the theory of invariance to reject the nuisance. Unfortunately, this can lead to a serious degradation of the detector capacity because some anomalies are masked by nuisance parameters. Nevertheless, in many cases the physical nature of nuisance parameters is (partially) known, and this a priori knowledge permits to define lower and upper bounds for the nuisance parameters. The goal of this paper is to study the statistical performances of the constrained generalized likelihood ratio test used to detect an additive anomaly in the case of bounded nuisance parameters. An example of the integrity monitoring of GNSS train positioning illustrates the relevance of the proposed method.