Random Distortion Testing and Optimality of Thresholding Tests

This paper addresses the problem of testing whether the Mahalanobis distance between a random signal $ \\Theta $ and a known deterministic model $ \\theta _{0}$ exceeds some given non-negative real number or not, when $ \\Theta $ has unknown probability distribution and is observed in additive independent Gaussian noise with positive definite covariance matrix. When $ \\Theta $ is deterministic unknown, we prove the existence of thresholding tests on the Mahalanobis distance to $ \\theta _{0}$ that have specified level and maximal constant power (MCP). The MCP property is a new optimality criterion involving Wald's notion of tests with uniformly best constant power ( UBCP) on ellipsoids for testing the mean of a normal distribution. When the signal is random with unknown distribution, constant power maximality extends to maximal constant conditional power (MCCP) and the thresholding tests on the Mahalanobis distance to $ \\theta _{0}$ still verify this novel optimality property. Our results apply to the detection of signals in independent and additive Gaussian noise. In particular, for a large class of possible model mismatches, MCCP tests can guarantee a specified false alarm probability, in contrast to standard Neyman-Pearson tests that may not respect this constraint.