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 $\sqrt{n}$-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.
翻译:标记依赖性在研究问题中起着关键作用, 这些问题可以被纳入标记进程的一般框架。 在这项工作中, 我们侧重于在估计标记进程的平均值和共变量功能时, 调整标记依赖性, 这是因为标记进程是独立复制标记进程。 我们假设标记进程是一个高斯进程, 点进程是一个日志- Gausian Cox 进程, 标记依赖性是通过两个潜潜伏高斯进程之间的依赖性产生的。 在这个框架内, 忽略标记依赖性的地方天真的线性估测器可能会有严重偏差。 我们表明, 可以用一个局部的跨差函数线性估计来纠正这一偏差, 并且建立偏差校正的估测器的统一趋同率。 此外, 我们提出一个基于当地标记独立线性估测器的测试数据, 这表明, 标记依赖性正常分布在一个准值 $\sqrt{n}- convergerence 比率中。 我们为关键模型假设设计了模型诊断工具, 并且用一个稳健的功能性测算法测试, 将提出一个更精确的普通的等级, 测试, 将提出一个模拟数据测试。