TY - JOUR
T1 - Weighted Frechet means as convex combinations in metric spaces: Properties and generalized median inequalities
AU - Ginestet, Cedric E.
AU - Simmons, Andrew
AU - Kolaczyk, Eric D.
PY - 2012/10
Y1 - 2012/10
N2 - 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. (C) 2012 Elsevier B.V. All rights reserved.
AB - 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. (C) 2012 Elsevier B.V. All rights reserved.
U2 - 10.1016/j.spl.2012.06.001
DO - 10.1016/j.spl.2012.06.001
M3 - Article
SN - 0167-7152
VL - 82
SP - 1859
EP - 1863
JO - Statistics & Probability Letters
JF - Statistics & Probability Letters
IS - 10
ER -