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Limiting behaviour of Fréchet means in the space of phylogenetic trees

Barden, D.; Le, Huiling; Owen, M.


D. Barden

Huiling Le

M. Owen


As demonstrated in our previous work on T4, the space of phylogenetic trees with four leaves, the topological structure of the space plays an important role in the non-classical limiting behaviour of the sample Fréchet means in T4. Nevertheless, the techniques used in that paper cannot be adapted to analyse Fréchet means in the space Tm of phylogenetic trees with m(⩾5)m(⩾5) leaves. To investigate the latter, this paper first studies the log map of Tm. Then, in terms of a modified version of this map, we characterise Fréchet means in Tm that lie in top-dimensional or co-dimension one strata. We derive the limiting distributions for the corresponding sample Fréchet means, generalising our previous results. In particular, the results show that, although they are related to the Gaussian distribution, the forms taken by the limiting distributions depend on the co-dimensions of the strata in which the Fréchet means lie.


Barden, D., Le, H., & Owen, M. (2018). Limiting behaviour of Fréchet means in the space of phylogenetic trees. Annals of the Institute of Statistical Mathematics, 70(1), 99-129.

Journal Article Type Article
Acceptance Date Sep 12, 2016
Online Publication Date Oct 19, 2016
Publication Date Feb 1, 2018
Deposit Date Nov 22, 2016
Publicly Available Date Nov 22, 2016
Journal Annals of the Institute of Statistical Mathematics
Print ISSN 0020-3157
Electronic ISSN 1572-9052
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 70
Issue 1
Pages 99-129
Keywords Central limit theorem, Fréchet mean, Log map, Phylogenetic trees, Stratified manifold
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Additional Information The final publication is available at Springer via


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