Article
Title: Uncertainty Quantification in Robust Inference for Irregularly Spaced Spatial Data Using Block Bootstrap
Author(s): S. N. Lahiri
Issue: Volume 80 Series A Part 3 Year 2018
Pages: 173 -- 221
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, . . . , sn, 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.