Variance Estimation for the General Regression Estimator

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ABSTRACT
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 dualpurpose
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 deleteone
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.
KEY WORDS: Confidence interval coverage; Hat matrix; Jackknife; Leverage;
Model unbiased; Skewness