{"paper":{"title":"Integral approximation by kernel smoothing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Bernard Delyon, Fran\\c{c}ois Portier","submitted_at":"2014-09-02T14:47:03Z","abstract_excerpt":"Let $(X_1,\\ldots,X_n)$ be an i.i.d. sequence of random variables in $\\mathbb{R}^d$, $d\\geq 1$. We show that, for any function $\\varphi :\\mathbb{R}^d\\rightarrow\\mathbb{R}$, under regularity conditions, \\[n^ {1/2}\\Biggl(n^{-1}\\sum_{i=1}^n\\frac{\\varphi(X_i)}{\\widehat{f}^(X_i)}- \\int \\varphi(x)\\,dx\\Biggr)\\stackrel{\\mathbb{P}}{\\longrightarrow}0,\\] where $\\widehat{f}$ is the classical kernel estimator of the density of $X_1$. This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\\widehat{f}=f$. Although this paper highlights some ap"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0733","kind":"arxiv","version":2},"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"}