In this short note, we study the properties of the weighted Frechet mean as a convex combination operator on an arbitrary metric space, (Y,d). We show that this binary operator is commutative, non-associative, idempotent, invariant to multiplication by a constant weight and possesses an identity element. We also treat the properties of the weighted cumulative Frechet mean. These tools allow us to derive several types of median inequalities for abstract metric spaces that hold for both negative and positive Alexandrov spaces. In particular, we show through an example that these bounds cannot be improved upon in general metric spaces. For weighted Frechet means, however, such inequalities can solely be derived for weights equal or greater than one. This latter limitation highlights the inherent difficulties associated with working with abstract-valued random variables.
Transformed Fréchet Means for Robust Estimation in Hadamard Spaces
Technical University of Munich（慕尼黑工业大学）；Munich Climate Center（慕尼黑气候中心）；TUM School of Engineering and Design, Department of Aerospace and Geodesy, Earth System Modelling Group（慕尼黑工业大学工程与设计学院，航空航天与大地测量系，地球系统建模组）；Potsdam Institute for Climate Impact Research（波茨坦气候影响研究所）；Artificial Intelligence Group（人工智能小组）
Uniform Convergence of Generalized Conditional Fréchet Means with Applications to Weighted Fréchet Aggregation and Exceedance Set Estimation
Research School of Finance, Actuarial Studies and Statistics, Australian National University（澳大利亚国立大学金融、精算与统计研究学院）；Department of Actuarial Studies and Business Analytics, Macquarie University（麦考瑞大学精算与商业分析系）
