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

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  • Ginestet, Cedric E.
  • Simmons, Andrew
  • Kolaczyk, Eric D.

Abstract

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.

Suggested Citation

  • Ginestet, Cedric E. & Simmons, Andrew & Kolaczyk, Eric D., 2012. "Weighted Frechet means as convex combinations in metric spaces: Properties and generalized median inequalities," Statistics & Probability Letters, Elsevier, vol. 82(10), pages 1859-1863.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:10:p:1859-1863
    DOI: 10.1016/j.spl.2012.06.001
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    References listed on IDEAS

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    1. David Balding & Pablo Ferrari & Ricardo Fraiman & Mariela Sued, 2009. "Limit theorems for sequences of random trees," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(2), pages 302-315, August.
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    Cited by:

    1. Antonio Irpino & Rosanna Verde, 2015. "Basic statistics for distributional symbolic variables: a new metric-based approach," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 9(2), pages 143-175, June.

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