A Generalized Multiplicity-adjusted Horvitz-Thompson Class of Multiple Frame Estimators

We consider the problem of combining different random samples from multiple frames such that the combined sample estimator is design unbiased for the target population covered by the union of all frames. Although this is the general multiple frame estimation problem, we address in this paper the specific case where we have only basic information about sampled units, i.e., selection probabilities from the frame unit is sampled from and the number of frames the unit belongs is known but identification of all such frames may not be available (except from the frame the unit was actually sampled from) due to sensitive information such as drug behavior or elusive behavior such as illegal immigration and ex-imprisonment. Having only basic information about frame membership of the sampled units renders all the MF-estimators unusable except for the one recently developed by Mecatti (2007) based on the simple idea of multiplicity estimators. We first propose a general formulation for MF-estimators, termed as the generalized Multiplicityadjusted Horvitz-Thompson (GMHT) estimators. We then propose a new estimator termed Hybrid Multiplicity which can use both basic information for some units and partial or full information for others, as the case may be.