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.强度函数的参数化建模，是空间点过程（spatial point processes）统计学领域的一项基础性研究课题。因此，针对给定的点模式数据集，评估所提出模型的适配性是至关重要的工作。为此，本文针对指定的参数化一阶强度函数（first-order intensity），构建了一类全新的半参数拟合优度检验方法，无需依赖现有主流蒙特卡洛（Monte-Carlo）检验所需的完整数据生成机制。所提出的检验方法核心依赖于对点过程二阶特性的精准非参数估计。为此，本文针对满足温和形状约束的聚类型空间点过程，提出了一种全新的非参数对相关函数（pair correlation function, PCF）估计量，经证明该估计量可实现一致相合性（uniform consistency）。所提出的检验统计量因具有闭式渐近分布（closed-form asymptotic distributions）而计算效率极高，且即便在检验复合假设（composite hypotheses）时仍可达到名义显著性水平（nominal size）。在实际应用中，本文提出的估计与检验流程可为参数化强度函数建模优化提供有效的工具支撑，这一点通过大量仿真实验以及对华盛顿哥伦比亚特区（Washington DC）街头犯罪活动的真实数据分析得到了验证。