Distribution of Increases in Residual Log Likelihood for Nested Spatial Models

Abstract

Here we review nested relationships between models in the Matérn family of spatial models. The problem of comparing nested statistical models is straightforward in regular parametric problems via the likelihood ratio statistics and its asymptotic distribution. Here we examine the distribution of increments in residual log likelihood between nested spatial models when the null hypothesis is that the spatial structure is a convex combination of white noise and the de Wijs process, also known by its logarithmic covariance function. This study is carried out by simulation of spatial processes and the important aspects of this work include how to simulate a spatial process of order 0, the lack of strong bias in the estimates of variance components, and the validity of the usual asymptotic results for nested spatial models examined here.

Keywords:

• Bias in REML estimates
• de Wijs process
• Likelihood ratio test
• Matérn covariance function

Mathematics Subject Classification:

• 62H10
• 62H11

Acknowledgment

The author wishes to thank Dr. Geoff Robinson for his comments during the preparation of this article. His attention has helped to clarify the results of this study. The work of the author was carried out when he was a graduate student in the Department of Statistics at the University of Chicago.