Testing First-Order Spherical Symmetry of Spatial Point Processes

Abstract

This article proposes a Kolmogorov-Smirnov-type test to assess spherical symmetry of the first-order intensity function of a spatial point process (SPP).Spherical symmetry, which is an important assumption in the well-known ETAS (epidemic type aftershock sequence) model, means that the intensity function of an SPP is invariant under a spherical transformation in an Euclidean space. An important property of first-order spherical symmetry is that the expected number of points within a sector region is proportional to the angle measure of the region.This provides a way to construct our test statistic. The asymptotic distribution of the test statistic is obtained under the framework of increasing domain asymptotics with weak dependence. We show that the resulting test statistic converges weakly to the absolute maximum of a zero mean Gaussian process under the null hypothesis and it is also consistent under the alternative hypothesis. A simulation study shows that the type I error probability of the test is close to the significance level, and the power increases to one as the magnitude of non-spherical symmetry increases.In an application of the ETAS model to Japan earthquakes, the article concludes that the first-order spherical symmetry assumption can be roughly accepted.