Uncertainty Quantification in Robust Inference for Irregularly Spaced Spatial Data Using Block Bootstrap

Abstract

This paper deals with uncertainty quantification (UQ) for a class of robust estimators of population parameters of a stationary, multivariate random field that is observed at a finite number of locations s₁,…, s_n, generated by a stochastic design. The class of robust estimators considered here is given by the so-called M-estimators that in particular include robust estimators of location, scale, linear regression parameters, as well as the maximum likelihood and pseudo maximum likelihood estimators, among others. Finding practically useful UQ measures, both in terms of standard errors of the point estimators as well as interval estimation for the parameters is a difficult problem due to presence of inhomogeneous dependence among irregularly spaced spatial observations. Exact and asymptotic variances of such estimators have a complicated form that depends on the autocovariance function of the random field, the spatial sampling density, and also on the relative rate of growth of the sample size versus the volume of the sampling region. Similar complex interactions of these factors are also present in the sampling distributions of these estimators which makes exact calibration of confidence intervals impractical. Here it is shown that a version of the spatial block bootstrap can be used to produce valid UQ measures, both in terms of estimation of the standard error as well as interval estimation. A key advantage of the proposed method is that it provides valid approximations in very general settings without requiring any explicit adjustments for spatial sampling structures and without requiring explicit estimation of the covariance function and of the spatial sampling density.