Gaussian approximation and spatially dependent wild bootstrap for high-dimensional spatial data

TL;DR

This paper develops a high-dimensional central limit theorem and a spatially dependent wild bootstrap method for irregularly spaced high-dimensional spatial data, enabling reliable statistical inference in complex spatial settings.

Contribution

It introduces a novel high-dimensional CLT and a spatially dependent wild bootstrap applicable to irregular spatial data, with theoretical validation and practical demonstrations.

Findings

Bootstrap method is asymptotically valid in high dimensions.

Method effectively constructs joint confidence intervals.

Useful for change-point detection in spatio-temporal data.

Abstract

In this paper, we establish a high-dimensional CLT for the sample mean of $p$-dimensional spatial data observed over irregularly spaced sampling sites in $\mathbb{R}^d$, allowing the dimension $p$ to be much larger than the sample size $n$. We adopt a stochastic sampling scheme that can generate irregularly spaced sampling sites in a flexible manner and include both pure increasing domain and mixed increasing domain frameworks. To facilitate statistical inference, we develop the spatially dependent wild bootstrap (SDWB) and justify its asymptotic validity in high dimensions by deriving error bounds that hold almost surely conditionally on the stochastic sampling sites. Our dependence conditions on the underlying random field cover a wide class of random fields such as Gaussian random fields and continuous autoregressive moving average random fields. Through numerical simulations and a real data analysis, we demonstrate the usefulness of our bootstrap-based inference in several applications, including joint confidence interval construction for high-dimensional spatial data and change-point detection for spatio-temporal data.