Testing the Mahalanobis distance between a random signal with unknown distribution and a known deterministic model in additive and independent standard Gaussian noise: the random distortion testing problem.
Dominique Pastor, Quang-Thang Nguyen · 2012
This paper addresses the problem of deciding whether the Mahalanobis distance between a random signalΘ and a knowndeterministicmodel θ0 exceeds some given non-negative real number or not, when the Θ has unknown probability distribution and is observed in additive independent Gaussian noise with positive definite covariance matrix. A new optimality criterion based on the invariance of both the problem and the noise distribution is introduced, via conditional notions of power and size. The tests optimal with respect to this criterion are given. The results established in this paper extend those of Wald’s for testing the mean of a Gaussian distribution, which is the particular case where τ= 0 and the signal is deterministic unknown. Application to the detection of random signals in additive independent Gaussian noise is also addressed.