  In this paper, the Gaussian quasi likelihood ratio test (GQLRT) for
non-Bayesian binary hypothesis testing is generalized by applying a transform
to the probability distribution of the data. The proposed generalization,
called measure-transformed GQLRT (MT-GQLRT), selects a Gaussian probability
model that best empirically fits a transformed probability measure of the data.
By judicious choice of the transform we show that, unlike the GQLRT, the
proposed test is resilient to outliers and involves higher-order statistical
moments leading to significant mitigation of the model mismatch effect on the
decision performance. Under some mild regularity conditions we show that the
MT-GQLRT is consistent and its corresponding test statistic is asymptotically
normal. A data driven procedure for optimal selection of the measure
transformation parameters is developed that maximizes an empirical estimate of
the asymptotic power given a fixed empirical asymptotic size. A Bayesian
extension of the proposed MT-GQLRT is also developed that is based on selection
of a Gaussian probability model that best empirically fits a transformed
conditional probability distribution of the data. In the Bayesian MT-GQLRT the
threshold and the measure transformation parameters are selected via joint
minimization of the empirical asymptotic Bayes risk. The non-Bayesian and
Bayesian MT-GQLRTs are applied to signal detection and classification, in
simulation examples that illustrate their advantages over the standard GQLRT
and other robust alternatives.
