Gaussian Approximation and Spatially Dependent Wild Bootstrap for High-Dimensional Spatial Data

Abstract

In this article, 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. Supplementary materials for this article are available online.

KEYWORDS:

• Change-point analysis
• High-dimensional central limit theorem
• Irregularly spaced spatial data
• Spatio-temporal data

Supplementary Materials

The supplement contains the high-dimensional CLT under polynomial moment condition (Appendix A), discussion on examples of random fields that satisfy our regularity conditions (Appendix B), some applications of SDWB for spatial and spatio-temporal data (Appendix C), proofs of Theorems 4.1 and A.1, and Corollary 4.1 (Appendix D), proof of Theorem 4.2 (Appendix E), proof of Proposition B.1 (Appendix F), proof of Proposition C.1 (Appendix G), technical tools (Appendix H), additional simulation results (Appendix I), and real data analysis (Appendix J).

Acknowledgments

Daisuke Kurisu is Associate Professor at The University of Tokyo, Japan. Kengo Kato is Professor of Statistics at Cornell University, and Xiaofeng Shao is Professor of Statistics at University of Illinois at Urbana-Champaign. We would like to thank three anonymous referees and an Associate Editor for constructive comments, which led to substantial improvements. We also would like to thank Adam Kashlak, Yuta Koike, Yasumasa Matsuda, Fabian Mies, Nan Zou, Taisuke Otsu and Yoshihiro Yajima for their helpful comments and suggestions.

Additional information

Funding

D. Kurisu is partially supported by JSPS KAKENHI Grant Number 20K13468. K. Kato is partially supported by NSF grants DMS-1952306, DMS-2014636, and DMS-2210368. X. Shao is partially supported by NSF grants DMS-1807032 and DMS-2014018.