Signal presence hypothesis testing with scaled covariance matrices

In order to maximize the signal detection probability, while maintaining a pre-specified false alarm rate, it is well known that the Neyman-Pearson likelihood ratio test (LRT) is the optimal sufficient statistic. As the distributions involved might depend on some unknown parameters, commonly a generalized LRT (GLRT) is evaluated where an estimate replaces the true value of the unknown parameters. It is usually assumed that for both hypotheses the noise level is unaltered. In practice though, this is rarely the case, as inaccurate modeling of the signal structure introduces additional errors. In this work we analyze the GLRT in the Gaussian setting assuming a scaled form of covariance matrices, which better describe practical scenarios. It is shown that by taking into account the difference of the noise level, it is possible to obtain an improved receiver operating characteristic (ROC) curve.