Regression Estimator (Survey)

The (generalized) regression estimator, often abbreviated GREG, estimates a population total using ${\hat{Y}}_{reg}=N\bar{y}+\hat{b}(X-N\bar{x})N$ (in mean form: ${\bar{y}}_{reg}=\bar{y}+\hat{b}({\mu }_x-\bar{x})$), where $\hat{b}$ is the estimated slope from regressing $y$ on the auxiliary variable $x$ within the sample and ${\mu }_x=X/N$ is the known population mean of $x$; the correction term adjusts the raw sample mean by the estimated regression relationship between the observed discrepancy in the auxiliary variable (${\mu }_x-\bar{x}$, how far the sample's auxiliary mean is from the known population value) and the outcome. Unlike the ratio estimator, which forces the relationship between $y$ and $x$ through the origin, the regression estimator allows a free intercept and is asymptotically at least as efficient as both the simple sample mean and the ratio estimator, reducing to the ordinary sample mean when $\hat{b}=0$ (no linear relationship) and generalizing further, in the case of multiple auxiliary variables via multiple regression, to the modern generalized regression (GREG) estimators widely used in official survey statistics for calibration.

• The regression estimator dominates the ratio estimator whenever the true relationship between y and x does not pass through the origin, since forcing a zero intercept in that situation (as the ratio estimator does) introduces avoidable bias that the regression estimator's free intercept corrects for.
• Modern national statistical agencies typically use generalized regression (GREG) estimators with multiple auxiliary variables simultaneously (age, region, prior administrative measures) rather than a single-variable regression estimator, extending the same underlying logic, use known population totals of auxiliary information to correct and stabilize the sample-based estimate.
• The regression estimator's calibration property, its estimated population total for the auxiliary variable itself exactly matches the known population total by construction, is what connects it directly to calibration estimation more broadly, GREG estimators are in fact one specific, widely used family within the broader calibration estimator framework.