Asymptotic theory of generalized information criterion for geostatistical regression model selection

TL;DR

This paper investigates the asymptotic properties of the generalized information criterion (GIC) for selecting geostatistical regression models within a mixed domain framework, providing theoretical guarantees and practical insights.

Contribution

It establishes the selection consistency and asymptotic loss efficiency of GIC under general conditions, even with covariance model misspecification, and explores the impact of domain growth and regressor smoothness.

Findings

GIC is selection consistent under certain conditions.

GIC may fail to identify the true polynomial order.

Domain growth and regressor smoothness influence model selection performance.

Abstract

Information criteria, such as Akaike's information criterion and Bayesian information criterion are often applied in model selection. However, their asymptotic behaviors for selecting geostatistical regression models have not been well studied, particularly under the fixed domain asymptotic framework with more and more data observed in a bounded fixed region. In this article, we study the generalized information criterion (GIC) for selecting geostatistical regression models under a more general mixed domain asymptotic framework. Via uniform convergence developments of some statistics, we establish the selection consistency and the asymptotic loss efficiency of GIC under some regularity conditions, regardless of whether the covariance model is correctly or wrongly specified. We further provide specific examples with different types of explanatory variables that satisfy the conditions. For example, in some situations, GIC is selection consistent, even when some spatial covariance parameters cannot be estimated consistently. On the other hand, GIC fails to select the true polynomial order consistently under the fixed domain asymptotic framework. Moreover, the growth rate of the domain and the degree of smoothness of candidate regressors in space are shown to play key roles for model selection.