MinMax sampling is a technique for downsampling a real-valued vector which
minimizes the maximum variance over all vector components. This approach is
useful for reducing the amount of data that must be sent over a constrained
network link (e.g. in the wide-area). MinMax can provide unbiased estimates of
the vector elements, along with unbiased estimates of aggregates when vectors
are combined from multiple locations. In this work, we propose a biased MinMax
estimation scheme, B-MinMax, which trades an increase in estimator bias for a
reduction in variance. We prove that when no aggregation is performed, B-MinMax
obtains a strictly lower MSE compared to the unbiased MinMax estimator. When
aggregation is required, B-MinMax is preferable when sample sizes are small or
the number of aggregated vectors is limited. Our experiments show that this
approach can substantially reduce the MSE for MinMax sampling in many practical
settings.

MinMax 采样是一种降采样实向量的技术，旨在最小化所有向量分量上的最大方差。本研究提出一种有偏的 MinMax 估算方案 (B-MinMax)，通过增加估算偏差来减小方差。当无聚合时，B-MinMax 的均方差严格低于无偏的 MinMax 估计器；当需要聚合时，在样本量较小或聚合向量数目有限时，B-MinMax 较为优选。实验证明，在实际应用中，该方法可以大幅降低 MinMax 采样的均方差。