Frequency domain bootstrap methods for random fields

Abstract

This paper develops a frequency domain bootstrap method for random fields on ${\mathbb{Z}}^2$. Three frequency domain bootstrap schemes are proposed to bootstrap Fourier coefficients of observations. Then, inverse-transformations are applied to obtain resamples in the spatial domain. As a main result, we establish the invariance principle of the bootstrap samples, from which it follows that the bootstrap samples preserve the correct second-order moment structure for a large class of random fields. The frequency domain bootstrap method is simple to apply and is demonstrated to be effective in various applications including constructing confidence intervals of correlograms for linear random fields, testing for signal presence using scan statistics, and testing for spatial isotropy in Gaussian random fields. Simulation studies are conducted to illustrate the finite sample performance of the proposed method and to compare with the existing spatial block bootstrap and subsampling methods.