Sketch-Based Estimation of Subpopulation-Weight
  Summaries of massive data sets support approximate query processing over the original data. A basic aggregate over a set of records is the weight of subpopulations specified as a predicate over records' attributes. Bottom-k sketches are a powerful summarization format of weighted items that includes priority sampling and the classic weighted sampling without replacement. They can be computed efficiently for many representations of the data including distributed databases and data streams.   We derive novel unbiased estimators and efficient confidence bounds for subpopulation weight. Our estimators and bounds are tailored by distinguishing between applications (such as data streams) where the total weight of the sketched set can be computed by the summarization algorithm without a significant use of additional resources, and applications (such as sketches of network neighborhoods) where this is not the case.   Our rigorous derivations are based on clever applications of the Horvitz-Thompson estimator, and are complemented by efficient computational methods. We demonstrate their benefit on a wide range of Pareto distributions. 
Edith Cohen and Haim Kaplan