标题： 新的M估计量的主成分 显示英文标题

标题： New M-estimator of the leading principal component

摘要： 我们研究非凸且不可微的目标函数$v \mapsto \mathrm{E} ( \| X - v \| \| X + v \| - \| X \|^2 )$在$\mathbb{R}^p$中的最小化问题。 特别是，我们证明其最小值点在特定条件下能够恢复椭圆对称$X$的第一主成分方向。 这些条件的严格性在各种情景中得到研究，包括变量数量发散的情况$p$。 我们建立了样本最小值点的一致性和渐近正态性。 我们提出了一种类似Weiszfeld的算法来优化目标函数，并证明它保证在有限步数内收敛。 结果通过两个模拟进行了说明。 显示英文摘要

摘要： We study the minimization of the non-convex and non-differentiable objective function $v \mapsto \mathrm{E} ( \| X - v \| \| X + v \| - \| X \|^2 )$ in $\mathbb{R}^p$. In particular, we show that its minimizers recover the first principal component direction of elliptically symmetric $X$ under specific conditions. The stringency of these conditions is studied in various scenarios, including a diverging number of variables $p$. We establish the consistency and asymptotic normality of the sample minimizer. We propose a Weiszfeld-type algorithm for optimizing the objective and show that it is guaranteed to converge in a finite number of steps. The results are illustrated with two simulations.