Semi-parametric goodness-of-fit test for clustered point processes with a shape-constrained pair correlation function

Specification of a parametric model for the intensity function is a fundamental task in statistics for spatial point processes. It is, therefore, crucial to be able to assess the appropriateness of a suggested model for a given point pattern data set. For this purpose, we develop a new class of semi-parametric goodness-of-fit tests for the specified parametric first-order intensity, without assuming a full data generating mechanism that is needed for the existing popular Monte-Carlo tests. The proposed tests crucially rely on accurate nonparametric estimation of the second-order properties of a point process. To address this we propose a new nonparametric pair correlation function (PCF) estimator for clustered spatial point processes under some mild shape constraints, which is shown to achieve uniform consistency. The proposed test statistics are computationally efficient owing to closed-form asymptotic distributions and achieve the nominal size even for testing composite hypotheses. In practice, the proposed estimation and testing procedures provide effective tools to improve parametric intensity function modeling, which is demonstrated through extensive simulation studies as well as a real data analysis of street crime activity in Washington DC.