This article studies the use of asymmetric loss functions for the optimal prediction of positive-valued spatial processes. We focus on the family of power-divergence loss functions due to its many convenient properties, such as its continuity, convexity, relationship to well known divergence measures, and the ability to control the asymmetry and behaviour of the loss function via a power parameter. The properties of power-divergence loss functions, optimal power-divergence (OPD) spatial predictors, and related measures of uncertainty quantification are examined. In addition, we examine the notion of asymmetry in loss functions defined for positive-valued spatial processes and define an asymmetry measure that is applied to the power-divergence loss function and other common loss functions. The paper concludes with a spatial statistical analysis of zinc measurements in the soil of a floodplain of the Meuse River, Netherlands, using OPD spatial prediction.

翻译：本文研究了非对称损失函数在正值空间过程最优预测中的应用。我们聚焦于幂散度损失函数族，因其具有诸多便利性质，包括连续性、凸性、与已知散度测度的关联性，以及通过幂参数调控损失函数非对称性与行为的能力。本文探讨了幂散度损失函数的性质、最优幂散度空间预测器及相关不确定性量化测度。此外，我们考察了针对正值空间过程定义的损失函数中非对称性的概念，并提出了一种非对称性度量方法，将其应用于幂散度损失函数及其他常见损失函数。最后，本文基于荷兰默兹河漫滩土壤锌浓度测量数据，展示了采用最优幂散度空间预测方法进行空间统计分析的应用案例。