Subsampling Methods to Estimate the Variance of Sample Means Based on Nonstationary Spatial Data With Varying Expected Values

Abstract

Subsampling and block resampling methods have been suggested in the literature to nonparametrically estimate the variance of statistics computed from spatial data. Usually stationary data are required. However, in empirical applications, the assumption of stationarity often must be rejected. This article proposes nonparametric methods to estimate the variance of (functions of) sample means based on nonstationary spatial data using subsampling. We assume that data are observed on a lattice in some region of R². In the data that we consider, the information in the different picture elements (pixels) of the lattice are allowed to come from different distributions, with smoothly varying expected values, or with expected values decomposed additively into directional components. Furthermore, pixels are assumed to be locally dependent, and the dependence structure is allowed to differ over the lattice. Consistent variance estimators for (functions of) sample means, together with convergence rates in mean square, are provided under these assumptions. An example with applications to forestry, using satellite data, is discussed.

KEY WORDS :

• Bootstrap
• Nonidentically distributed variables
• Nonindependent variables
• Resampling