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.

翻译：最大$L_q$似然估计器（ML$q$E）是一种参数估计方法，它作为最大似然估计器（MLE）的替代方案，考虑似然值的$q$次幂（其中$q<1$）。在此方法中，极端值因其较低的似然值而被降权，从而产生稳健的估计。本文研究了具有复现的空间数据ML$q$E的性质。我们探讨了具有Matérn协方差函数的高斯随机场ML$q$E的渐近性质，并通过模拟研究考察了ML$q$E的数值性能。结果表明，当空间数据中的部分复现包含异常值时，ML$q$E能够提供更稳健、更稳定的估计结果。此外，我们开发了一种机制来确定ML$q$E中超参数$q$的最优选择。我们方法的稳健性在一个美国降水数据集上得到了进一步验证。与其他针对空间数据的稳健方法相比，我们的方案更直观、更易于理解，并且在处理包含异常值的数据集时表现良好。