  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.
