A Weighted Likelihood Approach Based on Statistical Data Depths

TL;DR

This paper introduces a robust weighted likelihood estimation method using statistical data depths, which maintains efficiency like MLE in clean data and offers high robustness against contamination, applicable even in high-dimensional settings.

Contribution

It develops a novel weighted likelihood estimator based on data depths that is robust, efficient, and applicable in high-dimensional scenarios without requiring density smoothness.

Findings

Achieves asymptotic normality similar to MLE in clean data.

Maintains robustness in contaminated data.

Valid in high-dimensional settings without extra assumptions.

Abstract

We propose a general approach to construct weighted likelihood estimating equations with the aim of obtaining robust parameter estimates. We modify the standard likelihood equations by incorporating a weight that reflects the statistical depth of each data point relative to the model, as opposed to the sample. An observation is considered regular when the corresponding difference of these two depths is close to zero. When this difference is large the observation score contribution is downweighted. We study the asymptotic properties of the proposed estimator, including consistency and asymptotic normality, for a broad class of weight functions. In particular, we establish asymptotic normality under the standard regularity conditions typically assumed for the maximum likelihood estimator (MLE). Our weighted likelihood estimator achieves the same asymptotic efficiency as the MLE in the absence of contamination, while maintaining a high degree of robustness in contaminated settings. In stark contrast to the traditional minimum divergence/disparity estimators, our results hold even if the dimension of the data diverges with the sample size, without requiring additional assumptions on the existence or smoothness of the underlying densities. We also derive the finite sample breakdown point of our estimator for both location and scatter matrix in the elliptically symmetric model. Detailed results and examples are presented for robust parameter estimation in the multivariate normal model. Robustness is further illustrated using two real data sets and a Monte Carlo simulation study.