On the epsilon-optimal discrimination of two one-dimensional subspaces

This paper addresses the problem of discriminating two different vector lines from a non-zero mean Gaussian noise vector. Under each hypothesis, the Gaussian noise vector is completely characterized by its expected value which belongs to a known vector line. A new criterion of optimality, namely the epsilon most stringent test, is proposed and studied. This criterion consists in minimizing the maximum shortcoming of the test, up to a small loss, subject to a constrained false alarm probability. The maximum shortcoming corresponds to the maximum gap between the power function of the test and the envelope power function which is defined as the supremum of the power over all tests satisfying the prescribed false alarm probability. It is numerically shown that the proposed test outperforms the generalized likelihood ratio test.