AI 中文总结
本文为连续空间域上的两阶段抽样建立了基于设计的渐近理论，推导了Horvitz-Thompson估计量的方差阶数，证明了估计量的一致性及渐近正态性，并通过森林清查模拟验证了有限样本性能。
AI 中文摘要
本文针对连续空间域上的两阶段抽样，发展了一种基于设计的渐近理论。目标参数是积分总量，或此类总量的光滑函数，定义在一个固定的有界区域内，该区域被划分为越来越细的一级抽样单元集合。在此固定区域框架内，我们推导了当二级抽样单元是从连续子区域中选取的点时，Horvitz-Thompson估计量方差分量的阶数。我们在包含概率、抽样密度和研究变量的显式正则条件下，建立了估计量的设计一致性以及方差估计量的一致性。结果被推广到总量光滑尺度不变函数的插件估计量。对于高熵一级设计，我们进一步证明了仅基于一阶包含概率的Hájek型方差估计量是一致的，并证明了总量和插件估计量的渐近正态性。一项受森林清查应用启发的模拟研究展示了所提估计量和方差估计量的有限样本性能。
英文摘要
This paper develops a design-based asymptotic theory for two-stage sampling over a continuous spatial domain. The target parameters are integral totals, or smooth functions of such totals, defined over a fixed bounded territory partitioned into an increasingly fine collection of primary sampling units. Within this fixed-area framework, we derive the order of the variance components of the Horvitz-Thompson estimator when secondary sampling units are points selected from continuous sub-regions. We establish design consistency of the estimator, as well as consistency of variance estimators under explicit regularity conditions on inclusion probabilities, sampling densities, and the study variable. The results are extended to plug-in estimators of smooth scaleinvariant functions of totals. For high-entropy first-stage designs, we further show that a H{á}jek-type variance estimator based only on first-order inclusion probabilities is consistent, and we prove asymptotic normality of both total and plug-in estimators. A simulation study inspired by forest inventory applications illustrates the finite-sample performance of the proposed estimators and variance estimators.
