{"ID":2860839,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.02799","arxiv_id":"2510.02799","title":"New M-estimator of the leading principal component","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2510.02799","url_pdf":"https://arxiv.org/pdf/2510.02799v1","authors":"[\"Joni Virta\",\"Una Radojicic\",\"Marko Voutilainen\"]","published":"2025-10-03T08:13:45Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"stat.ME\"]","methods":"[]","has_code":false}