Variance Estimation for the General Regression Estimator

A variety of estimators of the variance of the general regression (GREG) estimator of a mean have been proposed in the sampling literature, mainly with the goal of estimating the design-based variance. Estimators can be easily constructed that, under certain conditions, are approximately unbiased for both the design-variance and the model-variance. Several dual-purpose estimators are studied here in single-stage sampling. These choices are robust estimators of a model-variance even if the model that motivates the GREG has an incorrect variance parameter. A key feature of the robust estimators is the adjustment of squared residuals by factors analogous to the leverages used in standard regression analysis. We also show that the delete-one jackknife implicitly includes the leverage adjustments and is a good choice from either the design-based or model-based perspective. In a set of simulations, these variance estimators have small bias and produce confidence intervals with near-nominal coverage rates for several sampling methods, sample sizes, and populations in single-stage sampling. We also present simulation results for a skewed population where all variance estimators perform poorly. Samples that do not adequately represent the units with large values lead to estimated means that are too small, variance estimates that are too small, and confidence intervals that cover at far less than the nominal rate. These defects need to be avoided at the design stage by selecting samples that cover the extreme units well. However, in populations with inadequate design information this will not be feasible.