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：我们提出了一种作用于泊松点过程的凸包算子，其最简单的例子是欧几里得空间中点过程支撑集的凸包。假设过程的强度度量在凸包算子生成的集合上已知，我们讨论了基于泊松过程构建的期望线性统计量的估计问题。在特殊情况下，我们的通用框架可得到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泊松过程的Kabanov-Skorohod积分给出。该方法的关键要素是底层泊松过程关于凸包的空间强马尔可夫性质。我们推导了估计误差的正态收敛速度，并将其应用于Hölder函数积分的估计量示例。此外，我们还探讨了高阶对称统计量的估计。