Robust Maximum $L_q$-Likelihood Covariance Estimation for Replicated Spatial Data

TL;DR

This paper introduces a robust covariance estimation method for replicated spatial data using the maximum $L_q$-likelihood estimator, demonstrating improved stability and robustness against outliers through theoretical analysis and empirical validation.

Contribution

It develops a new robust estimation approach for spatial data with replicates, including theoretical properties, optimal hyper-parameter selection, and practical application.

Findings

ML$q$E provides more robust estimates with outliers.

The method outperforms traditional approaches in simulations.

Application to US precipitation data confirms robustness.

Abstract

Parameter estimation with the maximum $L_q$-likelihood estimator (ML$q$E) is an alternative to the maximum likelihood estimator (MLE) that considers the $q$-th power of the likelihood values for some $q<1$. In this method, extreme values are down-weighted because of their lower likelihood values, which yields robust estimates. In this work, we study the properties of the ML$q$E for spatial data with replicates. We investigate the asymptotic properties of the ML$q$E for Gaussian random fields with a Mat\'ern covariance function, and carry out simulation studies to investigate the numerical performance of the ML$q$E. We show that it can provide more robust and stable estimation results when some of the replicates in the spatial data contain outliers. In addition, we develop a mechanism to find the optimal choice of the hyper-parameter $q$ for the ML$q$E. The robustness of our approach is further verified on a United States precipitation dataset. Compared with other robust methods for spatial data, our proposal is more intuitive and easier to understand, yet it performs well when dealing with datasets containing outliers.