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

TL;DR

This paper develops bias-corrected estimators and a test for mark-point dependence in marked point processes, improving inference accuracy and providing diagnostic tools through theoretical analysis and real data applications.

Contribution

It introduces a bias correction method for mark-point dependence and a test for independence within a Gaussian process framework, with proven convergence properties.

Findings

Bias correction reduces estimation bias significantly.

The independence test achieves asymptotic normality at $\

Simulation and real data demonstrate improved inference and model diagnostics.

Abstract

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.