  In this paper, we develop a generalization of the Gaussian quasi score test
(GQST) for composite binary hypothesis testing. The proposed test, called
measure transformed GQST (MT-GQST), is based on the score-function of the
measure transformed Gaussian quasi maximum likelihood estimator (MT-GQMLE) that
operates by empirically fitting a Gaussian model to a transformed probability
measure of the data. By judicious choice of the transform we show that, unlike
the GQST, the proposed MT-GQST involves higher-order statistical moments and
can gain resilience to outliers, leading to significant mitigation of the model
mismatch effect on the decision performance. A data-driven procedure for
optimal selection of the measure transformation parameters is developed that
minimizes the spectral norm of the empirical asymptotic error-covariance of the
MT-GQMLE. This amounts to maximization of an empirical worst-case asymptotic
local power at a fixed asymptotic size. The MT-GQST is applied to location
mismatch detection of a near-field point source in a simulation example that
illustrates its robustness to outliers.
