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Abstract:
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Under some regularity conditions, including that the process is Gaussian, the sampling region is rectangular, and that the parameter space $\Theta$ is compact, Matsuda and Yajima (2009) showed that the Whittle estimator $\widehat{\theta}_{n}$ minimizing their version of Whittle likelihood is consistent (for $d\leq 3$) and one can construct large sample confidence regions for covariance parameters $\theta$ using the asymptotic normality of the Whittle estimator $\widehat{\theta}_{n}$. However, this requires one to estimate the asymptotic covariance matrix, which involves integrals of the spatial sampling density. Moreover, nonparametric estimation of the quantities in the asymptotic covariance matrix requires specification of a smoothing parameter and is subject to the curse of dimensionality. In comparison, we propose a spatial frequency domain empirical likelihood method (cf. Bandyopadhyay et al. (2015), Van Hala et al. (2017)) based approach which can be employed to produce asymptotically valid confidence regions and tests on $\theta$, without requiring explicit estimation of such quantities.
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