Kriging Beyond Gaussianity: Fixed-Domain Asymptotic Optimality

Abstract

Under squared-error loss, the optimal predictor of a non-Gaussian random field is its conditional mean, which may be difficult to compute. We consider a broad class of non-Gaussian random fields obtained by transforming finitely many independent Gaussian fields sharing a common correlation function, and study when kriging entails no first-order loss in mean squared prediction error (MSPE) under fixed-domain asymptotics. We compare the observed-data conditional mean, kriging with the observed covariance, kriging with the latent correlation, and a latent-information oracle. These predictors satisfy a finite-sample MSPE ordering. Under conditions linking transformation regularity to the high-frequency decay of the latent spectral density, all four MSPEs are asymptotically equivalent. For radially nonincreasing spectral densities with polynomial tails, the required regularity order is independent of both the spatial dimension and the number of latent fields. Exponential correlation also yields finite-sample bounds. Examples include Weibull, skew-Gaussian, and copula-based fields, while numerical comparisons illustrate finite-sample differences. Counterexamples show why covariance structure and first-order transformation regularity alone are insufficient. Finally, standardized prediction errors converge to a Gaussian scale mixture, which is standard normal exactly when the transformation's gradient norm is almost surely constant.