Scoring rules are aimed at evaluation of the quality of predictions, but can also be used for estimation of parameters in statistical models. We propose estimating parameters of multivariate spatial models by maximising the average leave-one-out cross-validation score. This method, LOOS, thus optimises predictions instead of maximising the likelihood. The method allows for fast computations for Gaussian models with sparse precision matrices, such as spatial Markov models. It also makes it possible to tailor the estimator's robustness to outliers and their sensitivity to spatial variations of uncertainty through the choice of the scoring rule which is used in the maximisation. The effects of the choice of scoring rule which is used in LOOS are studied by simulation in terms of computation time, statistical efficiency, and robustness. Various popular scoring rules and a new scoring rule, the root score, are compared to maximum likelihood estimation. The results confirmed that for spatial Markov models the computation time for LOOS was much smaller than for maximum likelihood estimation. Furthermore, the standard deviations of parameter estimates were smaller for maximum likelihood estimation, although the differences often were small. The simulations also confirmed that the usage of a robust scoring rule results in robust LOOS estimates and that the robustness provides better predictive quality for spatial data with outliers. Finally, the new inference method was applied to ERA5 temperature reanalysis data for the contiguous United States and the average July temperature for the years 1940 to 2023, and this showed that the LOOS estimator provided parameter estimates that were more than a hundred times faster to compute compared to maximum-likelihood estimation, and resulted in a model with better predictive performance.

翻译：评分规则旨在评估预测质量，但也可用于统计模型中的参数估计。我们提出通过最大化平均留一交叉验证分数来估计多元空间模型的参数。这一称为LOOS的方法因此优化预测而非最大化似然函数。该方法能够对具有稀疏精度矩阵的高斯模型（如空间马尔可夫模型）进行快速计算。通过选择最大化过程中使用的评分规则，还可以定制估计量对异常值的鲁棒性及其对空间不确定性变化的敏感性。通过模拟研究，从计算时间、统计效率和鲁棒性三个方面分析了LOOS中所用评分规则选择的影响。将多种常用评分规则及新提出的根评分规则与最大似然估计进行了比较。结果证实，对于空间马尔可夫模型，LOOS的计算时间远小于最大似然估计。此外，最大似然估计的参数估计标准差更小，但差异通常较小。模拟还证实，使用鲁棒评分规则可产生鲁棒的LOOS估计，且这种鲁棒性能为含异常值的空间数据提供更好的预测质量。最后，将新推断方法应用于美国本土的ERA5温度再分析数据及1940年至2023年的七月平均温度，结果表明LOOS估计量提供的参数估计计算速度比最大似然估计快百倍以上，且所得模型具有更优的预测性能。