This estimator also belongs to a variety called calibration estimators, as the second term here “corrects” (or “calibrates”) the Horvitz-Thompson estimator for Y using known population totals for X.
Note that the estimator for Bd is based on sample data. When the sample size is small, this estimate may be unstable. To address this, one can pool the data over domains to produce a single The resulting modified direct estimator, known as the survey regression estimator, is expressed as follows:
Gershunskaya illustrated how this estimator might be applied to NCSES R&D data. Let Xim be the known population payroll in industry-type i and state be the Horvitz-Thompson estimate of payroll in industry-type i and state m, be the Horvitz-Thompson estimate of payroll, national total for industry-type i, and be the Horvitz-Thompson estimate of R&D funds, national total for industry-type i. Then, one can compute
using national data for industry i in the survey regression estimator to estimate
Gershunskaya pointed out that, although in the survey regression estimator is based on a larger sample, the effective sample size still equals the domain sample size. To see why that is so, one can rewrite the survey regression estimator as
which shows that the survey regression estimator is a sum of the fitted values from a regression model based on predictors from the domain of interest, and it has a bias correction from weighting the residuals again from a regression using data only from that domain. Therefore, the eff-