We study a broad class of models called semiparametric spatial point processes where the first-order intensity function contains both a parametric component and a nonparametric component. We propose a novel, spatial cross-fitting estimator of the parametric component based on random thinning, a common simulation technique in point processes. The proposed estimator is shown to be consistent and in many settings, asymptotically Normal. Also, 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. Next, we present a new spatial kernel regression estimator that can estimate the nonparametric component of the intensity function at the desired rates for inference. Despite the dependence induced by the point process, we show that our estimator can be computed using existing software for generalized partial linear models in i.i.d. settings. We conclude with a small simulation study and a re-analysis of the spatial distribution of rainforest trees.

翻译：我们研究一类称为半参数空间点过程的广泛模型，其一阶强度函数同时包含参数分量和非参数分量。我们提出了一种基于随机稀释（点过程中常用的模拟技术）的参数分量新型空间交叉拟合估计量。该估计量被证明具有一致性，并在许多设定下具有渐近正态性。此外，我们将独立同分布设定中的半参数效率下界概念推广至空间点过程，并证明若该过程为泊松过程，所提估计量可达到效率下界。接着，我们提出一种新的空间核回归估计量，能够以推断所需的速率估计强度函数的非参数分量。尽管点过程会引入依赖性，我们证明该估计量可利用独立同分布设定中广义部分线性模型的现有软件进行计算。最后，我们通过一项小型模拟研究和对雨林树木空间分布的重新分析完成论证。