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 (1995) and S<(a)double over dot>rndal (1992) 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 (1995) and S<(a)double over dot>rndal (1992). The simulation results on the properties of the proposed variance estimators relative to the empirical mean squared error are reported.