{"paper":{"title":"Asymptotic normality of the $L_k$-error of the Grenander estimator","license":"","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Hendrik P. Lopuha\\\"a, Vladimir N. Kulikov","submitted_at":"2006-02-11T13:40:14Z","abstract_excerpt":"We investigate the limit behavior of the $L_k$-distance between a decreasing density $f$ and its nonparametric maximum likelihood estimator $\\hat{f}_n$ for $k\\geq1$. Due to the inconsistency of $\\hat{f}_n$ at zero, the case $k=2.5$ turns out to be a kind of transition point. We extend asymptotic normality of the $L_1$-distance to the $L_k$-distance for $1\\leq k<2.5$, and obtain the analogous limiting result for a modification of the $L_k$-distance for $k\\geq2.5$. Since the $L_1$-distance is the area between $f$ and $\\hat{f}_n$, which is also the area between the inverse $g$ of $f$ and the more"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602244","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"}