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Limit Theorems for the Inductive Mean on Metric Trees

Part of: Special processes Limit theorems Mathematical biology in general

Published online by Cambridge University Press:  14 July 2016

Bojan Basrak
Bojan Basrak*
Affiliation:
University of Zagreb
*
∗Postal address: Mathematics Department, University of Zagreb, Bijenička 30, Zagreb, Croatia. Email address: bbasrak@math.hr
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Abstract

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For random variables with values on binary metric trees, the definition of the expected value can be generalized to the notion of a barycenter. To estimate the barycenter from tree-valued data, the so-called inductive mean is constructed recursively using the weighted interpolation between the current mean and a new data point. We show the strong consistency of the inductive mean, but also that it, somewhat peculiarly, converges towards the true barycenter with different rates, and asymptotic distributions depending on the small variations of the underlying distribution.

Keywords

Limit theoremmetric treeinductive meanconvergence in distributionLindley process

MSC classification

Secondary: 60F05: Central limit and other weak theorems 92B10: Taxonomy, cladistics, statistics 60K10: Applications (reliability, demand theory, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1136 - 1149
Copyright
Copyright © Applied Probability Trust 2010 

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