The Annals of Statistics

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

Wei-Liem Loh

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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.

Article information

Ann. Statist., Volume 43, Number 6 (2015), 2766-2794.

Received: March 2015
Revised: July 2015
First available in Project Euclid: 7 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M30: Spatial processes
Secondary: 62M40: Random fields; image analysis

Gaussian random field higher-order quadratic variation irregularly spaced data Matérn covariance smoothness estimation


Loh, Wei-Liem. Estimating the smoothness of a Gaussian random field from irregularly spaced data via higher-order quadratic variations. Ann. Statist. 43 (2015), no. 6, 2766--2794. doi:10.1214/15-AOS1365.

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