What you can do with Coordinated Samples

TL;DR

This paper provides a fundamental analysis of coordinated sampling, characterizing the types of queries for which desirable estimators exist and introducing variance competitiveness as a measure of estimator quality.

Contribution

It offers a precise characterization of estimable queries under coordinated sampling and introduces variance competitiveness to evaluate estimator performance across data.

Findings

Characterization of estimable queries with desirable properties.

Construction of variance competitive estimators for functions with unbiased nonnegative estimators.

Insights into the limits and potential of coordinated sampling.

Abstract

Sample coordination, where similar instances have similar samples, was proposed by statisticians four decades ago as a way to maximize overlap in repeated surveys. Coordinated sampling had been since used for summarizing massive data sets. The usefulness of a sampling scheme hinges on the scope and accuracy within which queries posed over the original data can be answered from the sample. We aim here to gain a fundamental understanding of the limits and potential of coordination. Our main result is a precise characterization, in terms of simple properties of the estimated function, of queries for which estimators with desirable properties exist. 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 {\em variance competitiveness}, which means that the expectation of the square on any data is not too 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.