  This paper develops empirical likelihood methodology for irregularly spaced
spatial data in the frequency domain. Unlike the frequency domain empirical
likelihood (FDEL) methodology for time series (on a regular grid), the
formulation of the spatial FDEL needs special care due to lack of the usual
orthogonality properties of the discrete Fourier transform for irregularly
spaced data and due to presence of nontrivial bias in the periodogram under
different spatial asymptotic structures. A spatial FDEL is formulated in the
paper taking into account the effects of these factors. The main results of the
paper show that Wilks' phenomenon holds for a scaled version of the logarithm
of the proposed empirical likelihood ratio statistic in the sense that it is
asymptotically distribution-free and has a chi-squared limit. As a result, the
proposed spatial FDEL method can be used to build nonparametric, asymptotically
correct confidence regions and tests for covariance parameters that are defined
through spectral estimating equations, for irregularly spaced spatial data. In
comparison to the more common studentization approach, a major advantage of our
method is that it does not require explicit estimation of the standard error of
an estimator, which is itself a very difficult problem as the asymptotic
variances of many common estimators depend on intricate interactions among
several population quantities, including the spectral density of the spatial
process, the spatial sampling density and the spatial asymptotic structure.
Results from a numerical study are also reported to illustrate the methodology
and its finite sample properties.
