On empirical likelihood methods for irregularly located spatial data

Abstract

Empirical likelihood (EL) is a statistical framework that allows for likelihood-type inference without explicit distributional assumptions. EL formulates a likelihood function nonparametrically, producing likelihood ratio statistics for use in constructing confidence intervals and performing tests. EL ratio statistics have properties analogous to parametric likelihood ratio statistics, such as chi-square limiting distributions. EL methods formulated for independent data typically fail when applied to dependent random processes. There are two primary ways to adapt EL to dependent data. One is through the use of data blocking techniques, where data blocking aims to capture the local dependence structure. The other general approach is through a data transformation to typically weaken the dependence structure, often involving analysis in the frequency domain. A variety of EL methods have been developed for time series, but the application of EL to spatial data, particularly irregularly located spatial data, has received far less attention. We investigate two different EL methods for irregularly located spatial data. The first is a blockwise EL method, which allows for inference on means, marginal distributions, and spatial regression parameters. The second is a frequency domain EL method, allowing various estimation and tests about spatial covariance structures. One primary challenge in investigating inference methods for irregularly located spatial data is the question of the asymptotic structure. In time series, the asymptotic context derives from an increasing number of observations over time, but in the spatial setting, there are two different drivers of the asymptotic regime: the rate of growth of the number of points within the sampling region and the rate of growth of the volume of the sampling region. The growth rate of the number of points may be proportional to or greater than the growth rate of the volume of the region. These differences often cause dramatic changes in the limiting distribution of spatial statistics. A further challenge in the frequency domain setting is that the irregular xxxvi spacing eliminates orthogonality properties of discrete Fourier transforms as typical for equispaced time series. For the blockwise EL method, we show that log EL ratio statistics have chi-square limiting distributions, provide results from a simulation study, and apply the method to a real data example in the context of spatial regression. For the frequency domain EL method, we also show log EL ratio statistics have chi-square limiting distributions, and we provide results from an extensive simulation study. In both cases, the EL methods are valid regardless of the exact type of spatial asymptotic structure or the concentrations of random sampling locations.