Measure-Transformed Gaussian Quasi Score Test in the Presence of Nuisance Parameters

In this paper, we extend the measure-transformed Gaussian quasi score test (MT-GQST) for the case where nuisance parameters are present. The proposed extension is based on a zero-expectation property of a partial Gaussian quasi score function under the transformed null distribution. The nuisance parameters are estimated under the null hypothesis via the measure-transformed Gaussian quasi MLE. In the paper, we analyze the effect of the probability measure-transformation on the asymptotic detection performance of the extended MT-GQST. This leads to a data-driven procedure for selection of the generating function of the considered transform, called MT-function, which, in practice, weights the data points. Furthermore, we provide conditions on the MT-function to ensure stability of the asymptotic false-alarm-rate in the presence of noisy outliers. The extended MT-GQST is applied for testing a vector parameter of interest comprising a noisy multivariate linear data model in the presence of nuisance parameters. Simulation study illustrates its advantages over other robust detectors.