In this paper the Gaussian quasi maximum likelihood estimator (GQMLE) isgeneralized by applying a transform to the probability distribution of thedata. The proposed estimator, called measure-transformed GQMLE (MT-GQMLE),minimizes the empirical Kullback-Leibler divergence between a transformedprobability distribution of the data and a hypothesized Gaussian probabilitymeasure. By judicious choice of the transform we show that, unlike the GQMLE,the proposed estimator can gain sensitivity to higher-order statistical momentsand resilience to outliers leading to significant mitigation of the modelmismatch effect on the estimates. Under some mild regularity conditions we showthat the MT-GQMLE is consistent, asymptotically normal and unbiased.Furthermore, we derive a necessary and sufficient condition for asymptoticefficiency. A data driven procedure for optimal selection of the measuretransformation parameters is developed that minimizes the trace of an empiricalestimate of the asymptotic mean-squared-error matrix. The MT-GQMLE is appliedto linear regression and source localization and numerical comparisonsillustrate its robustness and resilience to outliers.
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