Use of Zero Functions for Combining Information from Multiple Frames.

A class of generalized multiplicity-adjusted Horvitz–Thompson (GMHT) estimators was introduced by Singh and Mecatti (J. Official Stat. 27(4):633–650, 2011, JOS) to provide a unified and systematic approach to existing estimators via GMHT-Regression. The main purpose of this chapter is to present key observations that led to the development of the unified principled framework. These are based on the use of zero functions as predictors in regression which play a fundamental role in statistical estimation such as quasi-likelihood. The key observations are listed below. First, an optimal combination of two estimators of the same domain is equivalent to regression of the simple multiplicity-adjusted HT (i.e., average of the two estimators) on the zero function defined by the difference of two estimators. Second, there are two types of zero functions for each overlapping domain—one based on domain count estimates and the other based on domain total estimates. Some estimators use both types of zero functions as predictors in regression while others optimally combine first the two domain count estimates and then apply Hájek-type ratio adjustments for each sample to optimally estimated domain counts before regressing on domain total zero functions. Third, the regression need not be optimal as it can be based on a working covariance structure to obtain robust consistent estimates; this is in view of the observation that optimal regression estimators are typically unstable (i.e., with high coefficient of variation) for complex designs due to lack of adequate degrees of freedom for estimating regression parameters. Fourth, for any regression estimator, it may be better to use count zero functions via Hájek-adjustment first because the initial GMHT estimator may not be well correlated with count zero functions. Fifth, it might be preferable to use a suitable working covariance matrix than the optimal one in order to obtain a calibration form in addition to making the estimator more stable. Finally, sixth, the basic principles used in the unified framework make it possible to construct new improved estimators over other estimators in the literature. GMHT-Regression estimators are constructed as sums of contributions from each frame which allow for application of standard variance estimation techniques.