{"paper":{"title":"Minimax rates of entropy estimation on large alphabets via best polynomial approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.ST","stat.TH"],"primary_cat":"cs.IT","authors_text":"Pengkun Yang, Yihong Wu","submitted_at":"2014-07-01T19:51:11Z","abstract_excerpt":"Consider the problem of estimating the Shannon entropy of a distribution over $k$ elements from $n$ independent samples. We show that the minimax mean-square error is within universal multiplicative constant factors of $$\\Big(\\frac{k }{n \\log k}\\Big)^2 + \\frac{\\log^2 k}{n}$$ if $n$ exceeds a constant factor of $\\frac{k}{\\log k}$; otherwise there exists no consistent estimator. This refines the recent result of Valiant-Valiant \\cite{VV11} that the minimal sample size for consistent entropy estimation scales according to $\\Theta(\\frac{k}{\\log k})$. The apparatus of best polynomial approximation "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0381","kind":"arxiv","version":3},"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"}