Several well known estimators of finite population mean and its functions are investigated under some standard sampling designs. Such functions of mean include the variance, the correlation coefficient and the regression coefficient in the population as special cases. We compare the performance of these estimators under different sampling designs based on their asymptotic distributions. Equivalence classes of estimators under different sampling designs are constructed so that estimators in the same class have equivalent performance in terms of asymptotic mean squared errors (MSEs). Estimators in different equivalence classes are then compared under some superpopulations satisfying linear models. It is shown that the pseudo empirical likelihood (PEML) estimator of the population mean under simple random sampling without replacement (SRSWOR) has the lowest asymptotic MSE among all the estimators under different sampling designs considered in this paper. It is also shown that for the variance, the correlation coefficient and the regression coefficient of the population, the plug-in estimators based on the PEML estimator have the lowest asymptotic MSEs among all the estimators considered in this paper under SRSWOR. On the other hand, for any high entropy $\pi$PS (HE$\pi$PS) sampling design, which uses the auxiliary information, the plug-in estimators of those parameters based on the H\'ajek estimator have the lowest asymptotic MSEs among all the estimators considered in this paper.

翻译：在若干标准抽样设计下，研究了有限总体均值及其函数的几种经典估计量。此类均值函数包括总体方差、相关系数和回归系数等特例。我们基于这些估计量的渐近分布，比较了它们在不同抽样设计下的表现。构建了不同抽样设计下估计量的等价类，使得同一等价类中的估计量在渐近均方误差（MSE）方面具有等价表现。随后在满足线性模型的若干超总体下，比较了不同等价类中的估计量。结果表明，在简单随机不放回抽样（SRSWOR）下，总体均值的伪经验似然（PEML）估计量在本文考虑的所有不同抽样设计的估计量中具有最低的渐近MSE。同时，对于总体方差、相关系数和回归系数，基于PEML估计量的插入估计量在SRSWOR下同样具有本文所有估计量中最低的渐近MSE。另一方面，对于使用辅助信息的高熵πPS（HEπPS）抽样设计，基于Hájek估计量的这些参数的插入估计量在本文所有估计量中具有最低的渐近MSE。