Random Distortion Testing and Optimality of Thresholding Tests

• Published in: IEEE transactions on signal processing 2013-08, Vol.61 (16), p.4161-4171
• Main Authors: Pastor, Dominique, Nguyen, Quang-Thang
• Format: Article
• Language: English
• Subjects: Applied sciences; Detection, estimation, filtering, equalization, prediction; Event testing; Exact sciences and technology; hypothesis testing; Information, signal and communications theory; invariance; Mahalanobis norm; Normal distribution; random distortion testing; Signal and communications theory; Signal, noise; Studies; Telecommunications and information theory; test with maximal constant conditional power; test with maximal constant power; test with uniformly best constant power
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2013.2265680

This paper addresses the problem of testing whether the Mahalanobis distance between a random signal Θ and a known deterministic model θ 0 exceeds some given non-negative real number or not, when Θ has unknown probability distribution and is observed in additive independent Gaussian noise with positive definite covariance matrix. When Θ is deterministic unknown, we prove the existence of thresholding tests on the Mahalanobis distance to θ 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 θ 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.