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Title: Nonparametric bias-correction and test for mark-point dependence with replicated marked point processes Authors:  Yehua Li - University of California at Riverside (United States)
Yongtao Guan - University of Miami (United States)
Ganggang Xu - University of Miami (United States) [presenting]
Emma Jingfei Zhang - University of Miami (United States)
Abstract: Mark-point dependence plays a critical role in research problems fitting into the general framework of marked point processes. We focus on nonparametrically 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 but the biases can be corrected using a local linear estimator of the cross-covariance function. Uniform convergence rates of the bias-corrected estimators are established under mild conditions. Furthermore, we propose a formal testing procedure for mark-point dependence. The proposed test statistic, though based on nonparametric estimators, converges to an asymptotic normal distribution in a surprising parametric root-n convergence rate. The effectiveness of the proposed methods is demonstrated using extensive simulations and applications to two real data examples.