Bias-correction and Test for Mark-point Dependence with Replicated Marked Point Processes

Mark-point dependence plays a critical role in research problems that can be fitted into the general framework of marked point processes. In this work, we focus on adjusting for mark-point dependence when estimating the mean and covariance functions of the mark process, given independent replicates of the marked point process. We assume that the mark process is a Gaussian process and the point process is a log-Gaussian Cox process, where the mark-point dependence is generated through the dependence between two latent Gaussian processes. Under this framework, naive local linear estimators ignoring the mark-point dependence can be severely biased. We show that this bias can be corrected using a local linear estimator of the cross-covariance function and establish uniform convergence rates of the bias-corrected estimators. Furthermore, we propose a test statistic based on local linear estimators for mark-point independence, which is shown to converge to an asymptotic normal distribution in a parametric n-convergence rate. Model diagnostics tools are developed for key model assumptions and a robust functional permutation test is proposed for a more general class of mark-point processes. The effectiveness of the proposed methods is demonstrated using extensive simulations and applications to two real data examples.

标记点依赖在可归入标记点过程（marked point process）一般研究框架的诸多问题中发挥着关键作用。本研究聚焦于：在给定标记点过程的独立重复样本的前提下，估计标记过程的均值与协方差函数时，如何校正标记点依赖带来的偏倚。我们假设标记过程为高斯过程（Gaussian process），点过程为对数高斯Cox过程（log-Gaussian Cox process），且标记点依赖通过两个隐高斯过程之间的依赖关系生成。在此框架下，忽略标记点依赖的朴素局部线性估计量可能会产生严重偏倚。本文证明，可通过互协方差函数的局部线性估计量校正该偏倚，并确立了偏倚校正估计量的一致收敛速率。此外，本文提出了一种基于局部线性估计量的标记点独立性检验统计量，该统计量可在参数化n收敛速率下收敛至渐近正态分布。本文针对核心模型假设开发了模型诊断工具，并针对更一般的标记点过程类别提出了稳健泛函置换检验方法。本文通过大量模拟实验与两个真实数据集的应用案例，验证了所提方法的有效性。