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Topics in Nonparametric Statistics pp 227–239Cite as

On the CLT on Low Dimensional Stratified Spaces

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 74))

Abstract

Noncategorical observations, when regarded as points on a stratified space, lead to a nonparametric data analysis extending data analysis on manifolds. In particular, given a probability measure on a sample space with a manifold stratification, one may define the associated Fréchet function, Fréchet total variance, and Fréchet mean set. The sample counterparts of these parameters have a more nuanced asymptotic behaviors than in nonparametric data analysis on manifolds. This allows for the most inclusive data analysis known to date. Unlike the case of manifolds, Fréchet sample means on stratified spaces may stick to a lower dimensional stratum, a new dimension reduction phenomenon. The downside of stickiness is that it yields a less meaningful interpretation of the analysis. To compensate for this, an extrinsic data analysis, that is more sensitive to input data is suggested. In this paper one explores analysis of data on low dimensional stratified spaces, via simulations. An example of extrinsic analysis on phylogenetic tree data is also given.

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Acknowledgements

The first, third, and fourth authors thank NSF for awards DMS-1106935 and DMS-0805977. Thanks to Susan Holmes for providing access to her SAMSI-AoOD presentations and useful conversations at MBI. This work is part a group effort that started in the late 1990s with the study of asymptotics of extrinsic and intrinsic sample means and function estimation on arbitrary manifolds by the second and third authors and Zinoviy Landsman. We would also like to thank our colleagues on the working group of Data Analysis on Sample Spaces with a Manifold Stratification created by the third coauthor and Ezra Miller at SAMSI in 2010, and in particular, special thanks go to Thomas Hotz, Stephan Huckemann, and Steve Marron.

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Authors and Affiliations

  1. Texas Tech University, Lubbock, TX, USA

    Leif Ellingson

  2. Radboud University Nijmegen, Nijmegen, The Netherlands

    Harrie Hendriks

  3. Florida State University, Tallahassee, FL, USA

    Vic Patrangenaru & Paul San Valentin

Authors
  1. Leif Ellingson
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  2. Harrie Hendriks
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  3. Vic Patrangenaru
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  4. Paul San Valentin
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Corresponding author

Correspondence to Leif Ellingson .

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Editors and Affiliations

  1. Department of Statistics, The Pennsylvania State University, University Park, Pennsylvania, USA

    Michael G. Akritas

  2. Department of Statistics, North Carolina State University, Raleigh, North Carolina, USA

    S. N. Lahiri

  3. Department of Mathematics, University of California, San Diego, California, USA

    Dimitris N. Politis

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Ellingson, L., Hendriks, H., Patrangenaru, V., Valentin, P.S. (2014). On the CLT on Low Dimensional Stratified Spaces. In: Akritas, M., Lahiri, S., Politis, D. (eds) Topics in Nonparametric Statistics. Springer Proceedings in Mathematics & Statistics, vol 74. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0569-0_21

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