Estimation of the asymptotic variance of univariate and multivariate random fields and statistical inference

Abstract

Correlated random fields are a common way to model dependence structures in high-dimensional data, especially for data collected in imaging. One important parameter characterizing the degree of dependence is the asymptotic variance which adds up all autocovariances in the temporal and spatial domain. Especially, it arises in the standardization of test statistics based on partial sums of random fields and thus the construction of tests requires its estimation. In this paper we propose consistent estimators for this parameter for strictly stationary $\varphi $-mixing random fields with arbitrary dimension of the domain and taking values in a Euclidean space of arbitrary dimension, thus allowing for multivariate random fields. We establish consistency, provide central limit theorems and show that distributional approximations of related test statistics based on sample autocovariances of random fields can be obtained by the subsampling approach.

As in applications the spatial-temporal correlations are often quite local, such that a large number of autocovariances vanish or are negligible, we also investigate a thresholding approach where sample autocovariances of small magnitude are omitted. Extensive simulation studies show that the proposed estimators work well in practice and, when used to standardize image test statistics, can provide highly accurate image testing procedures. Having in mind automatized applications on a big data scale as arising in data science problems, these examinations also cover the proposed data-adaptive procedures to select method parameters.