Estimating the smoothness of a Gaussian random field from irregularly spaced data via higher-order quadratic variations

This article introduces a method for estimating the smoothness of a stationary, isotropic Gaussian random field from irregularly spaced data. This involves novel constructions of higher-order quadratic variations and the establishment of the corresponding fixed-domain asymptotic theory. In particular, we consider: (i) higher-order quadratic variations using nonequispaced line transect data, (ii) second-order quadratic variations from a sample of Gaussian random field observations taken along a smooth curve in ${\mathbb{R}}^2$, (iii) second-order quadratic variations based on deformed lattice data on ${\mathbb{R}}^2$. Smoothness estimators are proposed that are strongly consistent under mild assumptions. Simulations indicate that these estimators perform well for moderate sample sizes.