Spectral Density Estimation of Function-Valued Spatial Processes

TL;DR

This paper introduces a nonparametric kernel-based method for estimating the spectral density of function-valued spatial processes, applicable to irregular sampling schemes and providing optimal convergence rates.

Contribution

It develops a flexible spectral density estimator for Hilbert space-valued processes with theoretical guarantees, including asymptotic normality and optimal rates under various sampling schemes.

Findings

Estimator achieves minimax rates on regular grids.

Asymptotic normality established for Gaussian processes.

Applicable to irregularly sampled functional data.

Abstract

The spectral density function describes the second-order properties of a stationary stochastic process on $\mathbb{R}^d$. This paper considers the nonparametric estimation of the spectral density of a continuous-time stochastic process taking values in a separable Hilbert space. Our estimator is based on kernel smoothing and can be applied to a wide variety of spatial sampling schemes including those in which data are observed at irregular spatial locations. Thus, it finds immediate applications in Spatial Statistics, where irregularly sampled data naturally arise. The rates for the bias and variance of the estimator are obtained under general conditions in a mixed-domain asymptotic setting. When the data are observed on a regular grid, the optimal rate of the estimator matches the minimax rate for the class of covariance functions that decay according to a power law. The asymptotic normality of the spectral density estimator is also established under general conditions for Gaussian Hilbert-space valued processes. Finally, with a view towards practical applications the asymptotic results are specialized to the case of discretely-sampled functional data in a reproducing kernel Hilbert space.