Estimation of variance of the ratio estimator

A general class of estimators of the variance of the ratio estimator is considered, which includes two standard estimators v0 and v2 and approximates another estimator vH suggested by Royall & Eberhardt (1975). Asymptotic expansions for the variances and biases of the proposed estimators are obtained. Based on this we obtain optimal variance estimators in the class and compare the relative merits of three estimators v0, v1 and v2 without any model assumption. Under a simple regression model a more definite comparison of v0, v1 and v2 is made in terms of variance and bias.