单调估计的方差竞争性：收紧边界
Variance Competitiveness for Monotone Estimation: Tightening the Bounds
摘要：
随机样本被广泛用于汇总海量数据集并促进可扩展的分析。协调抽样，其中不同数据集的样本“共享”随机化，是一种强大的方法，有助于更准确地估计许多聚集和相似性措施。我们最近制定了一个模型{ \it单调估计问题}（MEP），它可以应用于协调抽样，投影在一个单一的项目。然后，MEP估计器可以用于估计协调样本上的总和聚合，例如距离。对于MEP，我们感兴趣的估计是无偏的和非负的。我们提出了{ \it方差竞争性}作为估计量的质量度量：对于每个数据向量，我们考虑通过无偏非负估计量在其上可达到的最小方差。然后，我们将估计量的竞争性定义为数据上平方期望与最小可能值的最大比值。我们还提出了一个一般结构的L $^*$估计，这是定义为任何MEP的非负无偏估计存在，并在最多4-竞争。我们的目标是在{ \em通用比}上获得更严格的界限，我们将其定义为可以为任何MEP获得的最小竞争比。我们得到的上限为3.375，提高了L $^*$估计的$4$的界。我们还建立了1.44的下限。通过构造特定MEP的{ \it最优竞争}估计量得到下界。该结构是独立的利益，因为它有利于估计与实例最优的竞争力。
英文摘要：
Random samples are extensively used to summarize massive data sets and facilitate scalable analytics. Coordinated sampling, where samples of different data sets "share" the randomization, is a powerful method which facilitates more accurate estimation of many aggregates and similarity measures. We recently formulated a model of {\it Monotone Estimation Problems} (MEP), which can be applied to coordinated sampling, projected on a single item. MEP estimators can then be used to estimate sum aggregates, such as distances, over coordinated samples. For MEP, we are interested in estimators that are unbiased and nonnegative. We proposed {\it variance competitiveness} as a quality measure of estimators: For each data vector, we consider the minimum variance attainable on it by an unbiased and nonnegative estimator. We then define the competitiveness of an estimator as the maximum ratio, over data, of the expectation of the square to the minimum possible. We also presented a general construction of the L$^*$ estimator, which is defined for any MEP for which a nonnegative unbiased estimator exists, and is at most 4-competitive. Our aim here is to obtain tighter bounds on the {\em universal ratio}, which we define to be the smallest competitive ratio that can be obtained for any MEP. We obtain an upper bound of 3.375, improving over the bound of $4$ of the L$^*$ estimator. We also establish a lower bound of 1.44. The lower bound is obtained by constructing the {\it optimally competitive} estimator for particular MEPs. The construction is of independent interest, as it facilitates estimation with instance-optimal competitiveness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
