Effective sample size for spatial regression models

Abstract

We propose a new definition of effective sample size. Although the recent works of Griffith (2005, 2008) and Vallejos and Osorio (2014) provide a theoretical framework to address the reduction of information in a spatial sample due to spatial autocorrelation, the asymptotic properties of the estimations have not been studied in those studies or in previously ones. In addition, the concept of effective sample size has been developed primarily for spatial regression processes with a constant mean. This paper introduces a new definition of effective sample size for general spatial regression models that is coherent with previous definitions. The asymptotic normality of the maximum likelihood estimation is obtained under an increasing domain framework. In particular, the conditions for which the limiting distribution holds are established for the Matérn covariance family. Illustrative examples accompany the discussion of the limiting results, including some cases where the asymptotic variance has a closed form. The asymptotic normality leads to an approximate hypothesis testing that establishes whether there is redundant information in the sample. Simulation results support the theoretical findings and provide information about the behavior of the power of the suggested test. A real dataset in which a transect sampling scheme has been used is analyzed to estimate the effective sample size when a spatial linear regression model is assumed.