Variance estimation for the regression estimator in two-phase sampling

Many techniques in survey sampling depend on the possession of information about an auxiliary variable x, or a vector of auxiliary variables, available for the entire population. Regression estimates require (X) over bar, the population mean. If such information is unavailable, then one can sometimes obtain a large preliminary sample of zi relatively cheaply and use this to obtain a good estimate. say <(x)over bar '>, of (X) over bar. A smaller subsample can then be taken and the characteristic of interest, y(i), measured. A regression estimator can then be used treating <(x)over bar '> as if it were (X) over bar. This is termed double sampling, or two-phase sampling. This article focuses on variance estimators for the regression estimator in the aforementioned context and their use in constructing confidence intervals. A design-based linearization variance estimator that makes more complete use of the sample data than the standard one is considered for two-phase sampling. A jackknife variance estimator and its linearized version are obtained and shown to be design consistent. A bootstrap variance estimator is also shown to be design consistent. Unconditional and conditional repeated sampling properties of these variance estimators are studied through simulation. It is shown that the linearization variance estimator displays superior unconditional properties, but the jackknife ana its linearized version perform better conditionally.