On the Estimation of Integrated Covariance Functions of Stationary Random Fields

Abstract

For stationary vector-valued random fields on the asymptotic covariance matrix for estimators of the mean vector can be given by integrated covariance functions. To construct asymptotic confidence intervals and significance tests for the mean vector, non-parametric estimators of these integrated covariance functions are required. Integrability conditions are derived under which the estimators of the covariance matrix are mean-square consistent. For random fields induced by stationary Boolean models with convex grains, these conditions are expressed by sufficient assumptions on the grain distribution. Performance issues are discussed by means of numerical examples for Gaussian random fields and the intrinsic volume densities of planar Boolean models with uniformly bounded grains. Copyright (c) 2009 Board of the Foundation of the Scandinavian Journal of Statistics.

Central limit theorems for the excursion set volumes of weakly dependent random fields

“…A major disadvantage is tedious calculation in the case of a large observation window. The same problem arises for an estimator based on the covariance function estimation for the underlying random field; see [18] (cf. [12], Chapter 4).…”

“…One feasible estimator for the asymptotic covariance matrix Σ that arose in the multivariate CLT, see Theorem 2, can be called a subwindow estimator [18] and is constructed as follows. Let (V n ) n∈N and (W n ) n∈N be sequences of VH-growing sets (not necessarily rectangles) such that V n ⊂ W n , n ≥ 1.…”

Statistics for Gaussian random fields with unknown location and scale using Lipschitz‐Killing curvatures

“…We introduce a technique to estimate the asymptotic variances appearing in Propositions 2 and 3. Inspired by the cutting of T ( N ) introduced in Pantle et al (2010) (see also Bulinski et al, 2012, section 5) we consider a classical empirical variance estimator. To establish its consistency, we decompose this estimator on domains that are infinitely distant, mimicking the classical context of independent and identically distributed random variables.…”

“…These results permit to derive estimators and test for the underlying field X (see Berzin, 2018; Biermé et al, 2019; Di Bernardino et al, 2017). Inference methods and tests using LK curvature devices only rely on the sparse observation of one excursion set and not on the covariance function nor on the marginal distribution of X , which require the observation of the entire field (see, e.g., Nieto‐Reyes et al, 2014; Pantle et al, 2010).…”

Simultaneous confidence band for stationary covariance function of dense functional data

“…, Diggle and Verbyla (1998), Hall et al (1994), Yang et al (2016) and Yin et al (2010). In this sense, Cao et al (2016) proposed a simultaneous confidence envelope of covariance function for functional data; Horváth et al (2013) proposed a consistent estimator for the long-run covariance operator of stationary time series; Pantle et al (2010) considered the estimation of integrated covariance functions, which is required to construct asymptotic confidence intervals and significance tests for the mean vector in the context of stationary random fields. Since the covariance function measures stronger association among variables that are closer to each other, the employment of covariance function is considerably highlighted in spatial data analysis when the geometric structure of the surface is rough and self-similar.…”