Signal norm testing in additive and independent standard Gaussian noise

This paper addresses signal norm testing (SNT), that is, the problem of deciding whether a random signal norm exceeds some specified value τ > 0 or not, when the signal has unknown probability distribution and is observed in additive and independent standard Gaussian noise. The theoretical framework proposed for SNT extends usual notions in statistical inference and introduces a new optimality criterion. This one takes the invariance of both the problem and the noise distribution into account, via conditional notions of power and size and, more specifically, the introduction of the spherically-conditioned power function. The theoretical results established with respect to this criterion extend those deriving from standard statistical inference theory in the case of an unknown deterministic signal. Thinkable applications are problems where signal amplitude deviations from some nominal referencemust be detected above a certain tolerance τ, possibly chosen by the user on the basis of his experience and know-how. In this respect, the theoretical results of this paper are applied to an SNT formulation for the problem of detecting random signals in noise, with a specific focus on the case where the noise standard deviation is unknown.