Stationarity tests for spatial point processes using discrepancies

For testing stationarity of a given spatial point pattern, Guan (2008) proposed a model-free statistic, based on the deviations between observed and expected counts of points in expanding regions within the sampling window. This article extends his method to a general class of statistics by incorporating also such information when points are projected to the axes and by allowing different ways to construct regions in which the deviations are considered. The limiting distributions of the new statistics can be expressed in terms of integrals of a Brownian sheet and hence asymptotic critical values can be approximated. A simulation study shows that the new tests are always more powerful than that of Guan. When applied to the longleaf pine data where Guan's test gave an inconclusive answer, the new tests indicate a clear rejection of the stationarity hypothesis.