Nonparametric estimation of the pair correlation function of replicated inhomogeneous point processes

Abstract

We consider the nonparametric estimation of the isotropic pair correlation function (PCF) of inhomogeneous point processes when replicates are available. Based on carefully designed estimating equations, two types of nonparametric estimators, i.e., the local polynomial estimator and the orthogonal series estimator, are proposed and studied. The proposed estimators circumvent the problems caused by the need for estimating the unknown intensity function for kernel smoothed PCF estimators and they are free of edge correction terms. Asymptotic properties are investigated for both estimators and valid point-wise confidence bands are derived. Finite sample performances of the proposed estimators are demonstrated by simulation as well as an application to the Sina Weibo posting data.

Second-Order Semi-Parametric Inference for Multivariate Log Gaussian Cox Processes

“…They were, however, not able to obtain estimates of the individual cross pair correlation functions. Xu et al (2020) obtained a consistent estimate of a common pair correlation function in the presence of a common completely unspecified intensity function but they considered the particular multivariate setting of independent and identically distributed point processes.…”