Fixed-domain asymptotic properties of tapered maximum likelihood estimators

TL;DR

This paper examines how covariance tapering influences the asymptotic efficiency of maximum likelihood estimators for the Matérn covariance function in large spatial datasets, demonstrating conditions where tapered MLEs remain efficient.

Contribution

It provides a theoretical analysis of the fixed-domain asymptotic properties of tapered MLEs, establishing their efficiency under certain tapering conditions.

Findings

Tapered MLEs can be asymptotically as efficient as true MLEs.

Conditions on the taper ensure no loss of efficiency.

Results are specific to data collected along a line in a bounded region.

Abstract

When the spatial sample size is extremely large, which occurs in many environmental and ecological studies, operations on the large covariance matrix are a numerical challenge. Covariance tapering is a technique to alleviate the numerical challenges. Under the assumption that data are collected along a line in a bounded region, we investigate how the tapering affects the asymptotic efficiency of the maximum likelihood estimator (MLE) for the microergodic parameter in the Mat\'ern covariance function by establishing the fixed-domain asymptotic distribution of the exact MLE and that of the tapered MLE. Our results imply that, under some conditions on the taper, the tapered MLE is asymptotically as efficient as the true MLE for the microergodic parameter in the Mat\'ern model.