Statistical analysis of irregularly spaced spatial data in frequency domain

Abstract

Central limit theorems (CLTs) for frequency‐domain statistics are fundamental tools in frequency‐domain analysis. However, for irregularly spaced data, they are still limited. In both the pure increasing domain and the mixed increasing domain asymptotic frameworks, three CLTs of frequency‐domain statistics are established for the observations at uniformly distributed sampling locations over a rectangular sampling region. One is for discrete Fourier transforms (DFTs), while the other two pertain to generalized spectral means (GSMs). The asymptotic joint normality and independence of the DFT at any finite number of standard frequencies are derived. Additionally, the asymptotic normalities of two GSMs are set up, with asymptotic variances given in different forms, according to the Gaussian or non‐Gaussian model assumption. Three established CLTs are very useful in investigating the sampling properties of many important frequency‐domain statistics, such as periodogram, non‐negative definite auto‐covariance estimator, spectral density estimator, and Whittle likelihood estimator as well.