Statistica Sinica 14(2004), 69-101

BOOTSTRAPPING A SPATIAL POINT PROCESS

J. M. Loh and M. L. Stein

Columbia University and University of Chicago

Abstract: We consider the problem of resampling or bootstrapping a point process to get confidence intervals for the reduced second moment function. We propose a resampling scheme for spatial data, which we call the marked point method. This is a variant of the block of blocks bootstrap first introduced by Künsch (1989). A simulation study with a Poisson, a clustered and a regular point process on the unit square in ${\Bbb R}^2$ shows that the marked point method yields confidence intervals that are closer to the nominal (95%) level than resampling by tiling (block bootstrap) and by using subsets (subsampling). The confidence intervals obtained by the marked point method also tend to be shorter, after accounting for differences in empirical coverage. Finally, the marked point method is very much computationally less intensive so that, even with moderate sample sizes, the marked point method takes considerably less computing time. We also find that the simple method of dividing the sample and treating the subsamples as independent replicates works reasonably well. We apply some of these methods to a set of astronomy data.

Key words and phrases: Marked point bootstrap, reduced second moment function, resampling.