Binary Hypothesis Testing via Measure Transformed Quasi-Likelihood Ratio Test

Abstract

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.