Weighted Frechet means as convex combinations in metric spaces: Properties and generalized median inequalities

作者：Ginestet Cedric E^*; Simmons Andrew; Kolaczyk Eric D

来源：Statistics & Probability Letters, 2012, 82(10): 1859-1863.

DOI：10.1016/j.spl.2012.06.001

• Abstract-valued random variable
• Barycenter
• Convex operator
• Frechet mean
• Median inequality

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 cover 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 to or greater than one. This latter limitation highlights the inherent difficulties associated with abstract-valued random variables.

• 出版日期2012-10