{"paper":{"title":"Asymptotically Efficient Estimation of Smooth Functionals of Covariance Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Vladimir Koltchinskii","submitted_at":"2017-10-25T04:36:53Z","abstract_excerpt":"Let $X$ be a centered Gaussian random variable in a separable Hilbert space ${\\mathbb H}$ with covariance operator $\\Sigma.$ We study a problem of estimation of a smooth functional of $\\Sigma$ based on a sample $X_1,\\dots ,X_n$ of $n$ independent observations of $X.$ More specifically, we are interested in functionals of the form $\\langle f(\\Sigma), B\\rangle,$ where $f:{\\mathbb R}\\mapsto {\\mathbb R}$ is a smooth function and $B$ is a nuclear operator in ${\\mathbb H}.$ We prove concentration and normal approximation bounds for plug-in estimator $\\langle f(\\hat \\Sigma),B\\rangle,$ $\\hat \\Sigma:=n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.09072","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"}