Borrowing Strength Is Not the Best Technique Within a Wide Class of Design-Consistent Domain Estimators
Victor M. Estevao and Carl-Erik Särndal
Estimation for subpopulations, or domains, is an important objective in most surveys, especially in large surveys conducted by national statistical agencies. These agencies practice design-based domain estimation whenever possible, that is, whenever the sample size is sufficient and auxiliary information is available. The precision, as measured by the design-based variance, is a function of these factors. Insufficient precision - leading to a suppression of estimates - is more likely to happen for minor domains than for major domains. Our starting point is a statement of the auxiliary information available for a survey. Strong information provides the material for precise domain estimates. We form a class of domain estimators based on the given auxiliary information. It includes regression fit estimators as well as calibration estimators, direct as well as indirect estimators. Direct estimators use only y-values from inside the domain itself. Indirect estimators borrow strength by incorporating external y-values thought to be related. Borrowing strength is the cornerstone of small area estimation, a research tradition that is model-dependent, nondesign-based, and not examined in this article. The concept of borrowing strength is highly useful in that theory. However, since design-based domain estimation is extensively practiced, we are led to the question: What can borrowing strength do for design-based domain estimation? The answer is that borrowing strength is unfruitful in the design-based tradition. We find that for a fixed set of auxiliary information, the minimum asymptotic design-based variance is obtained with a direct estimator, derived by calibration rather than by regression fitting.
Design-based inference, very nearly design unbiased estimation, calibration, calibrated weights, regression fit, regression residuals