{"paper":{"title":"On the sample covariance matrix estimator of reduced effective rank population matrices, with applications to fPCA","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ME","stat.TH"],"primary_cat":"math.ST","authors_text":"Florentina Bunea, Luo Xiao","submitted_at":"2012-12-21T03:13:35Z","abstract_excerpt":"This work provides a unified analysis of the properties of the sample covariance matrix $\\Sigma_n$ over the class of $p\\times p$ population covariance matrices $\\Sigma$ of reduced effective rank $r_e(\\Sigma)$. This class includes scaled factor models and covariance matrices with decaying spectrum. We consider $r_e(\\Sigma)$ as a measure of matrix complexity, and obtain sharp minimax rates on the operator and Frobenius norm of $\\Sigma_n-\\Sigma$, as a function of $r_e(\\Sigma)$ and $\\|\\Sigma\\|_2$, the operator norm of $\\Sigma$. With guidelines offered by the optimal rates, we define classes of mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5321","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}