  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.
