On variance estimation of the ratio estimator under two-phase sampling

Using the estimated loss function approach, we compare the variance estimators of the ratio estimator proposed by Rao and Sitter () and Särndal () with the usual one under two-phase sampling. Our results show that their variance estimators are better than the usual one when the size of the second phase sample is greater than 2, which gives a theoretical verification of Rao and Sitter's observation through simulation. We also propose some other variance estimators which improve the ones in Rao and Sitter () and Särndal (). The simulation results on the properties of the proposed variance estimators relative to the empirical mean squared error are reported.