Abstract This article is concerned with random objects in the complex projective space ℂ P k − 2 . It is shown that the Veronese–Whitney (VW) antimean, which is the extrinsic antimean of a random point on ℂ P k − 2 relative to the VW-embedding, is given by the point on ℂ P k − 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 χ 2 -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.

中文翻译：

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平面 Kendall 形状空间上外在反均值和 Veronese-Whitney 均值和反均值估计的中心极限定理

摘要 本文关注复射影空间 ℂ P k − 2 中的随机对象。结果表明，Veronese-Whitney (VW) 反均值是 ℂ P k − 2 上随机点相对于​​ VW 嵌入的外在反均值，由 ℂ P k − 2 上的点给出，由特征向量表示对应于随机点的 VW 嵌入的预期平均值的最小特征值，前提是该特征值很简单。我们还导出了外在样本反均值的 CLT，以及基于外在反均值的适当学生化统计量的渐近 χ 2 分布，然后在 VW 嵌入的特定情况下用于构建非参数自举置信区域VW-antimean 平面 Kendall 形状。