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

TL;DR

This paper presents a novel method for estimating the smoothness of Gaussian random fields from irregular data using higher-order quadratic variations, with proven consistency and good performance in simulations.

Contribution

It introduces new constructions of higher-order quadratic variations for irregularly spaced data and establishes their asymptotic properties for smoothness estimation.

Findings

Estimators are strongly consistent under mild conditions.

Simulation results show good performance for moderate sample sizes.

Applicable to various data configurations like line transects and deformed lattices.

Abstract

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.