Smoothness estimation of nonstationary Gaussian random fields from irregularly spaced data observed along a curve

Abstract

This article considers estimating the smoothness parameter of a class of nonstationary Gaussian random fields on ${\mathbb{R}}^d$ using irregularly spaced data observed along a curve. The set of covariance functions includes a nonstationary version of the Matérn covariance function as well as isotropic Matérn covariance function. Smoothness estimators are constructed via higher-order quadratic variations. Under mild conditions, these estimators are shown to be strongly consistent and convergence rate upper bounds are established with respect to fixed-domain asymptotics. Simulations indicate that the proposed estimators perform well for moderate sample sizes.