论文标题
通过放松
Fréchet Mean Set Estimation in the Hausdorff Metric, via Relaxation
论文作者
论文摘要
这项工作解决了非欧几里得统计数据中的以下问题：对于Hausdorff Metric，是否有可能始终估计未知人口分布的平均值集，当访问独立的相同分布样本时？我们的肯定答案是基于对“放松的经验式平均估计量”的仔细分析，该分析识别经验Fréchet功能的一组近距离二聚体，而随着数据数量倾向于无限，“放松”的数量消失了。从理论方面来说，我们的结果包括确切的描述，其放松速率具有弱的一致性和较强的一致性，以及对估计器的描述（假设某些时刻的有限性和在基础度量空间的度量熵上的有限性和轻度条件）适应性地发现了最快的放松率，以实现强大一致的估计。在应用方面，我们考虑了估计具有热带投射度量标准的等距树空间中未知分布的Fermat-weber点集的问题；在这种情况下，我们提供了一种算法，可证明可以实现我们的适应性估计量，并将这种方法应用于实际系统发育数据。
This work resolves the following question in non-Euclidean statistics: Is it possible to consistently estimate the Fréchet mean set of an unknown population distribution, with respect to the Hausdorff metric, when given access to independent identically-distributed samples? Our affirmative answer is based on a careful analysis of the "relaxed empirical Fréchet mean set estimators" which identify the set of near-minimizers of the empirical Fréchet functional and where the amount of "relaxation" vanishes as the number of data tends to infinity. On the theoretical side, our results include exact descriptions of which relaxation rates give weak consistency and which give strong consistency, as well as a description of an estimator which (assuming only the finiteness of certain moments and a mild condition on the metric entropy of the underlying metric space) adaptively finds the fastest possible relaxation rate for strongly consistent estimation. On the applied side, we consider the problem of estimating the set of Fermat-Weber points of an unknown distribution in the space of equidistant trees endowed with the tropical projective metric; in this setting, we provide an algorithm that provably implements our adaptive estimator, and we apply this method to real phylogenetic data.
