Get the most out of your sample: Optimal unbiased estimators using partial information

Random sampling is an essential tool in the processing and transmission of data. It is used to summarize data too large to store or manipulate and meet resource constraints on bandwidth or battery power. Estimators that are applied to the sample facilitate fast approximate processing of queries posed over the original data and the value of the sample hinges on the quality of these estimators. Our work targets data sets such as request and traffic logs and sensor measurements, where data is repeatedly collected over multiple instances: time periods, locations, or snapshots. We are interested in operations, like quantiles and range, that span multiple instances. Subset-sums of these operations are used for applications ranging from planning to anomaly and change detection. Unbiased low-variance estimators are particularly effective as the relative error decreases with aggregation. The Horvitz-Thompson estimator, known to minimize variance for subset-sums over a sample of a single instance, is not optimal for multi-instance operations because it fails to exploit samples which provide partial information on the estimated quantity. We present a general principled methodology for the derivation of optimal unbiased estimators over sampled instances and aim to understand its potential. We demonstrate significant improvement in estimate accuracy of fundamental queries for common sampling schemes.