Measure-transformed quasi maximum likelihood estimation with application to source localization

In this paper, we consider the problem of estimating a deterministic vector parameter when the likelihood function is unknown or not expressible. We develop an estimator, called measure-transformed quasi maximum likelihood estimator (MT-QMLE), that minimizes the empirical Kullback-Leibler divergence between the transformed probability measure of the data and a hypothesized Gaussian probability distribution. By judicious choice of the transform we show that the proposed estimator can gain sensitivity to higher-order statistical information and resilience to outliers. Under some regularity conditions we show that the MT-QMLE is consistent, asymptotically normal and unbiased. Furthermore, we derive a necessary and sufficient condition for its asymptotic efficiency. The MT-QMLE is applied to source localization in a simulation example that illustrates its sensitivity to higher-order information and resilience to outliers.