We introduce a hull operator on Poisson point processes, the easiest example being the convex hull of the support of a point process in Euclidean space. Assuming that the intensity measure of the process is known on the set generated by the hull operator, we discuss estimation of an expected linear statistic built on the Poisson process. In special cases, our general scheme yields an estimator of the volume of a convex body or an estimator of an integral of a H\"older function. We show that the estimation error is given by the Kabanov--Skorohod integral with respect to the underlying Poisson process. A crucial ingredient of our approach is a spatial strong Markov property of the underlying Poisson process with respect to the hull. We derive the rate of normal convergence for the estimation error, and illustrate it on an application to estimators of integrals of a H\"older function. We also discuss estimation of higher order symmetric statistics.

翻译：我们引入Poisson点过程上的凸包算子，其中最简实例为欧氏空间中点过程支撑集的凸包。假设凸包算子生成集合上的过程强度测度已知，本文讨论了基于该Poisson过程构建的期望线性统计量的估计问题。在特例情形下，本通用框架可导出凸体体积的估计量或Hölder函数积分的估计量。我们证明估计误差由关于底层Poisson过程的Kabanov--Skorohod积分给出。分析方法的关键在于底层Poisson过程关于凸包的空间强马尔可夫性。我们推导了估计误差的正态收敛速率，并通过Hölder函数积分估计器应用实例进行说明，同时探讨了高阶对称统计量的估计问题。