Efficient estimation of semiparametric spatial point processes with V-fold random thinning

Abstract: We study a broad class of models called semiparametric spatial point processes where the intensity function contains both a parametric component and a nonparametric component. We propose a novel estimator of the parametric component based on random thinning, a common sampling technique in point processes. The proposed estimator of the parametric component is shown to be consistent and asymptotically normal if the nonparametric component can be estimated at the desired rate. We then extend a popular kernel-based estimator in i.i.d. settings and establish convergence rates that will enable inference for the parametric component. Next, we generalize the notion of semiparametric efficiency lower bound in i.i.d. settings to spatial point processes and show that the proposed estimator achieves the efficiency lower bound if the process is Poisson. Computationally, we show how to efficiently evaluate the proposed estimator with existing software for generalized partial linear models in i.i.d. settings by tailoring the sampling weights to replicate the dependence induced by the point process. We conclude with a small simulation study and a re-analysis of the spatial distribution of rainforest trees.