Measure-Transformed Quasi Maximum Likelihood Estimation

TL;DR

This paper introduces a measure-transformed Gaussian quasi maximum likelihood estimator (MT-GQMLE) that enhances robustness to outliers and sensitivity to higher-order moments by applying a distribution transform, improving estimation accuracy under model mismatch.

Contribution

The paper proposes a novel measure-transformed GQMLE that generalizes the traditional GQMLE, offering improved robustness and sensitivity through a data-driven transform parameter selection.

Findings

MT-GQMLE is consistent, asymptotically normal, and unbiased.

It demonstrates robustness to outliers in numerical experiments.

The method improves estimation accuracy in linear regression and source localization.

Abstract

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