What You Can Do with Coordinated Samples

Sample coordination, where similar instances have similar samples, was proposed by statisticiansfour decades ago as a way to maximize overlap in repeated surveys. Coordinated sampling had beensince used for summarizing massive data sets.The usefulness of a sampling scheme hinges on the scope and accuracy within which queries posedover the original data can be answered from the sample. We aim here to gain a fundamental understand-ing of the limits and potential of coordination. Our main result is a precise characterization, in termsof simple properties of the estimated function, of queries for which estimators with desirable propertiesexist. We consider unbiasedness, nonnegativity, finite variance, and bounded estimates.Since generally a single estimator can not be optimal (minimize variance simultaneously) for all data,we propose variance competitiveness, which means that the expectation of the square on any data is nottoo far from the minimum one possible for the data. Surprisingly perhaps, we show how to construct,for any function for which an unbiased nonnegative estimator exists, a variance competitive estimator. 1