{"paper":{"title":"A law of the iterated logarithm for Grenander's estimator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Jon A. Wellner, Lutz Duembgen, Malcolm Wolff","submitted_at":"2015-02-01T22:02:59Z","abstract_excerpt":"In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If $f(t_0) > 0$, $f'(t_0) < 0$, and $f'$ is continuous in a neighborhood of $t_0$, then \\begin{eqnarray*} \\limsup_{n\\rightarrow \\infty} \\left ( \\frac{n}{2\\log \\log n} \\right )^{1/3} ( \\widehat{f}_n (t_0 ) - f(t_0) ) = \\left| f(t_0) f'(t_0)/2 \\right|^{1/3} 2M \\end{eqnarray*} almost surely where $ M \\equiv \\sup_{g \\in {\\cal G}} T_g = (3/4)^{1/3}$ and $ T_g \\equiv \\mbox{argmax}_u \\{ g(u) - u^2 \\} $; here ${\\cal G}$ is the two-sided Strassen limit set on $R$. The proof re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00320","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"}