On False Alarm Rate of Matched Filter Under Distribution Mismatch

The generalized likelihood ratio test (GLRT) is a very widely used technique for detecting signals of interest amongst noise, when some of the parameters describing the signal (and possibly the noise) are unknown. The threshold of such a test is set from a desired probability of false alarm Pfa and hence this threshold depends on the statistical assumptions made about noise. In practice however, the noise statistics are seldom known and it becomes crucial to characterize Pfa under a mismatched distribution. In this letter, we address this problem in the case of a simple binary composite hypothesis testing problem (matched filter) when the threshold is designed under a Gaussian assumption while the noise actually follows an elliptically contoured distribution. We also consider the inverse situation. Generic expressions for the assumed and actual Pfa are derived and illustrated on the particular case of Student distributions for which simple, closed-form expressions are obtained. The latter show that the GLRT based on Gaussian assumption is not robust while that based on Student assumption is.