  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.
