{"paper":{"title":"On the acceleration of some empirical means with application to nonparametric regression","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Bernard Delyon (IRMAR), Fran\\c{c}ois Portier (IRMAR)","submitted_at":"2013-12-16T20:15:21Z","abstract_excerpt":"Let $(X_1,\\ldots ,X_n)$ be an i.i.d. sequence of random variables in $\\R^d$, $d\\geq 1$, for some function $\\varphi:\\R^d\\r \\R$, under regularity conditions, we show that \\begin{align*} n^{1/2} \\left(n^{-1} \\sum_{i=1}^n \\frac{\\varphi(X_i)}{\\w f^{(i)}(X_i)}-\\int_{} \\varphi(x)dx \\right) \\overset{\\P}{\\lr} 0, \\end{align*} where $\\w f^{(i)}$ is the classical leave-one-out 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 $\\w f^{(i)}=f$. As a consequence, it improves the classical Monte Car"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.4497","kind":"arxiv","version":1},"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"}