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.
Highlights
This paper introduces a new definition of effective sample size for spatial regression processes
Extensions of the Bayesian information criteria (BIC) for non-iid vectors require a definition of effective sample size that applies in such cases (Berger et al, 2014)
On the basis of model (1.2), we first provide a new definition of effective sample size as a function of the correlation structure of a process that does not depend on the scale of the variables
Summary
This paper introduces a new definition of effective sample size for spatial regression processes. On the basis of model (1.2), we first provide a new definition of effective sample size as a function of the correlation structure of a process that does not depend on the scale of the variables. The conditions given by Mardia and Marshall (1984) to obtain the asymptotic distribution of the ML estimator of θ are established, and the limiting distribution of the ESS is obtained using the delta method These conditions are particularized for a Gaussian random field with the Matern covariance, as well as for CAR and directional CAR processes. The programs that were used to analyze the data and the dataset can be obtained from http://spatialpack.mat.utfsm.cl
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