Random samples are lossy summaries which allow queries posed over the data to be approximated by applying an appropriate estimator to the sample. The effectiveness of sampling, however, hinges on estimator selection. The choice of estimators is subjected to global requirements, such as unbiasedness and range restrictions on the estimate value, and ideally, we seek estimators that are both efficient to derive and apply and admissible (not dominated, in terms of variance, by other estimators). Nevertheless, for a given data domain, sampling scheme, and query, there are many admissible estimators.
We define monotone sampling, which is implicit in many applications of massive data set analysis, and study the choice of admissible nonnegative and unbiased estimators. Our main contribution is general derivations of admissible estimators with desirable properties. We present a construction of order-optimal estimators, which minimize variance according to any specified priorities over the data domain. Order-optimality allows us to customize the derivation to common patterns that we can learn or observe in the data. When we prioritize lower values (e.g., more similar data sets when estimating difference), we obtain the L* estimator, which is the unique monotone admissible estimator and dominates the classic Horvitz-Thompson estimator. We show that the L* estimator is 4-competitive, meaning that the expectation of the square, on any data, is at most 4 times the minimum possible for that data. These properties make the L* estimator a natural default choice. We also present the U* estimator, which prioritizes large values (e.g., less similar data sets). Our estimator constructions are general, natural, and practical, allowing us to make the most from our summarized data.