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A Central Limit Theorem for extrinsic antimeans and estimation of Veronese–Whitney means and antimeans on planar Kendall shape spaces

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  • Wang, Yunfan
  • Patrangenaru, Vic
  • Guo, Ruite

Abstract

This article is concerned with random objects in the complex projective space ℂPk−2. It is shown that the Veronese–Whitney (VW) antimean, which is the extrinsic antimean of a random point on ℂPk−2 relative to the VW-embedding, is given by the point on ℂPk−2 represented by the eigenvector corresponding to the smallest eigenvalue of the expected mean of the VW-embedding of the random point, provided this eigenvalue is simple. We also derive a CLT for extrinsic sample antimeans, and an asymptotic χ2-distribution of an appropriately studentized statistic, based on the extrinsic antimean, which in the particular case of a VW-embedding is then used to construct nonparametric bootstrap confidence regions for the VW-antimean planar Kendall shape. Simulations studies and an application to medical imaging are illustrating the proposed methodology.

Suggested Citation

  • Wang, Yunfan & Patrangenaru, Vic & Guo, Ruite, 2020. "A Central Limit Theorem for extrinsic antimeans and estimation of Veronese–Whitney means and antimeans on planar Kendall shape spaces," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:jmvana:v:178:y:2020:i:c:s0047259x19301484
    DOI: 10.1016/j.jmva.2020.104600
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    1. Hendriks, Harrie & Landsman, Zinoviy, 1998. "Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 227-243, November.
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    1. Stephan F. Huckemann, 2021. "Comments on: Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 71-75, March.
    2. Vic Patrangenaru & Yifang Deng, 2021. "Extrinsic Regression and Anti-Regression on Projective Shape Manifolds," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 629-646, June.
    3. Ruite Guo & Hwiyoung Lee & Vic Patrangenaru, 2023. "Test for Homogeneity of Random Objects on Manifolds with Applications to Biological Shape Analysis," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1178-1204, August.

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