The Horvitz-Thompson (HT), the Rao-Hartley-Cochran (RHC) and the generalized regression (GREG) estimators of the finite population mean are considered, when the observations are from an infinite dimensional space. We compare these estimators based on their asymptotic distributions under some commonly used sampling designs and some superpopulations satisfying linear regression models. We show that the GREG estimator is asymptotically at least as efficient as any of the other two estimators under different sampling designs considered in this paper. Further, we show that the use of some well known sampling designs utilizing auxiliary information may have an adverse effect on the performance of the GREG estimator, when the degree of heteroscedasticity present in linear regression models is not very large. On the other hand, the use of those sampling designs improves the performance of this estimator, when the degree of heteroscedasticity present in linear regression models is large. We develop methods for determining the degree of heteroscedasticity, which in turn determines the choice of appropriate sampling design to be used with the GREG estimator. We also investigate the consistency of the covariance operators of the above estimators. We carry out some numerical studies using real and synthetic data, and our theoretical results are supported by the results obtained from those numerical studies.

翻译：本文考虑观测数据来自无限维空间时，有限总体均值的Horvitz-Thompson（HT）估计量、Rao-Hartley-Cochran（RHC）估计量以及广义回归（GREG）估计量。我们在若干常用抽样设计及满足线性回归模型的超总体条件下，基于这些估计量的渐近分布进行比较。研究表明，在本文考察的不同抽样设计下，GREG估计量渐近效率至少不低于其他两种估计量。进一步发现，当线性回归模型中异方差程度较小时，利用辅助信息的某些著名抽样设计可能对GREG估计量的表现产生不利影响；而当异方差程度较大时，采用这些抽样设计反而会提升该估计量的性能。我们开发了判定异方差程度的方法，该方法进而决定了与GREG估计量配合使用的合适抽样设计选择。我们还探讨了上述估计量协方差算子的一致性。通过真实数据与合成数据的数值实验，理论结果得到了数值结果的充分验证。