Testing proportionality between the first-order intensity functions of spatial point processes

This article proposes a Kolmogorov-Smirnov type test for proportionality between the first-order intensity functions of two independent spatial point processes. After appropriate scaling, the test statistic is constructed by maximizing the absolute difference between their point densities over a π-system. By treating non-stationary point processes as transformed from stationary point processes such that all questions of asymptotics related to the tightness can be answered, the article shows that the resulting test statistic converges weakly to the absolute maximum of a pinned Brownian sheet. This may be reduced to the standard Brownian bridge in a special case. A simulation study shows that the type I error probability of the test is close to the significance level and the power function increases to 1 as the magnitude of non-proportionality increases. In applications to two typical natural hazard data, the article concludes that the first-order intensity functions might be proportional in one case and not in the other.