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This paper develops empirical likelihood methodology for irregularly spacedspatial data in the frequency domain. Unlike the frequency domain empiricallikelihood (FDEL) methodology for time series (on a regular grid), theformulation of the spatial FDEL needs special care due to lack of the usualorthogonality properties of the discrete Fourier transform for irregularlyspaced data and due to presence of nontrivial bias in the periodogram underdifferent spatial asymptotic structures. A spatial FDEL is formulated in thepaper taking into account the effects of these factors. The main results of thepaper show that Wilks' phenomenon holds for a scaled version of the logarithmof the proposed empirical likelihood ratio statistic in the sense that it isasymptotically distribution-free and has a chi-squared limit. As a result, theproposed spatial FDEL method can be used to build nonparametric, asymptoticallycorrect confidence regions and tests for covariance parameters that are definedthrough spectral estimating equations, for irregularly spaced spatial data. Incomparison to the more common studentization approach, a major advantage of ourmethod is that it does not require explicit estimation of the standard error ofan estimator, which is itself a very difficult problem as the asymptoticvariances of many common estimators depend on intricate interactions amongseveral population quantities, including the spectral density of the spatialprocess, the spatial sampling density and the spatial asymptotic structure.Results from a numerical study are also reported to illustrate the methodologyand its finite sample properties.

A frequency domain empirical likelihood method for irregularly spaced spatial data