Random Distortion testing and optimality of thresholding testes

• Published in: IEEE transactions on signal processing Vol. 61; no. 16; pp. 4161 - 4171
• Main Authors: Pastor, Dominique, Nguyen, Quang Thang
• Format: Journal Article
• Language: English
• Published: Institute of Electrical and Electronics Engineers 01.08.2013
• Subjects: Engineering Sciences; Signal and Image processing
• ISSN: 1053-587X

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 mistmatches, \mccp~tests can guarantee a specified false alarm probability, in contrast to standard Neyman-Pearson tests that may not respect this constraint.