Consistency of bootstrap approximation to the null distributions of local spatial statistics with application to house price analysis

Chang-Lin Mei,Shou-Fang Xu,Feng Chen

doi:10.1186/s13660-020-02482-x

Abstract

With the increasing availability of spatially extensive geo-referenced data, much attention has been paid to the use of local statistics to identify local patterns of spatial association, in which the null distributions of local statistics play an essential role in the related statistical inference. As a powerful tool to approximate the distribution of a statistic, the bootstrap method is used in this paper to derive null distributions of the commonly used local spatial statistics including local Getis and Ord’s G_{i}, Moran’s I_{i} and Geary’s c_{i}. Strong consistency of the bootstrap approximation to the null distributions of the statistics is proved under some mild conditions, and the Boston housing price data are analyzed to demonstrate the application of the theoretical results.

Highlights

Exploration of spatial association has long been recognized as an important issue in spatial data analysis
In order to test for significance of local spatial association at a reference location, it is essential to derive the null distribution of the local statistics
The Kolmogorov distance is mainly used to investigate the strong consistency of the bootstrap approximation for the local Getis and Ord’s Gi, Moran’s Ii and Geary’s ci and the main results are summarized in the following theorems

Summary

Introduction

Exploration of spatial association has long been recognized as an important issue in spatial data analysis. Normal distributions have been used to approximate the null distributions of some local spatial statistics such as local Getis and Ord’s Gi, Moran’s Ii and Geary’s ci (see, for example, [1, 11, 18]). Yan et al [26] suggested a bootstrap method, originally proposed by Efron [8], to approximate the null distributions of the spatio-temporal versions of local Getis and Ord’s Gi, Moran’s Ii and Geary’s ci. The main objective of this paper is to theoretically investigate the validity of the bootstrap approximation to the null distributions of local Getis and Ord’s Gi, Moran’s Ii and Geary’s ci. The Kolmogorov distance is mainly used to investigate the strong consistency of the bootstrap approximation for the local Getis and Ord’s Gi, Moran’s Ii and Geary’s ci and the main results are summarized in the following theorems. Sup P∗ c∗i (d) ≤ x – P ci(d) ≤ x a.s

Preliminaries and lemmas

Proofs of the theorems

Wi2nε2

Findings

Monte Carlo implementation of the bootstrap distribution functions