Semiparametric estimation of spectral density function for irregular spatial data

TL;DR

This paper introduces a semi-parametric method for estimating the spectral density of isotropic spatial processes with irregular data, combining smoothing splines and parametric models to improve accuracy and theoretical justification.

Contribution

A novel semi-parametric spectral density estimator that effectively captures low and high frequency properties with theoretical backing and superior performance.

Findings

Outperforms existing non-parametric estimators in simulations

Provides asymptotic bounds for bias and variance

Effectively estimates spectral densities with irregular spatial data

Abstract

Estimation of the covariance structure of spatial processes is of fundamental importance in spatial statistics. In the literature, several non-parametric and semi-parametric methods have been developed to estimate the covariance structure based on the spectral representation of covariance functions. However,they either ignore the high frequency properties of the spectral density, which are essential to determine the performance of interpolation procedures such as Kriging, or lack of theoretical justification. We propose a new semi-parametric method to estimate spectral densities of isotropic spatial processes with irregular observations. The spectral density function at low frequencies is estimated using smoothing spline, while a parametric model is used for the spectral density at high frequencies, and the parameters are estimated by a method-of-moment approach based on empirical variograms at small lags. We derive the asymptotic bounds for bias and variance of the proposed estimator. The simulation study shows that our method outperforms the existing non-parametric estimator by several performance criteria.