{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:AA44WS24BRYC4E6R6G5JYWTFFZ","short_pith_number":"pith:AA44WS24","schema_version":"1.0","canonical_sha256":"0039cb4b5c0c702e13d1f1ba9c5a652e4e203351ab0d1fa46b1170005c7f9a21","source":{"kind":"arxiv","id":"1502.00320","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1502.00320","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2015-02-01T22:02:59Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"b948ff9cc7b1c1d6acc398a4aca2601912a1a379e6beb9a9a7afa25cb9be4f4b","abstract_canon_sha256":"acf26d229b45a165d70f13567ae70d31847e20c471036a11a7020dca1ba33b09"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:34.514005Z","signature_b64":"Dtfk5MO6vSV9zwAaTPTvOxhdsUk1ag1nWlRmHlLgJVtXzAuLUzAn8kLGeuJb4ZoClpf9hg/0SmzAjKcJMJQjCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0039cb4b5c0c702e13d1f1ba9c5a652e4e203351ab0d1fa46b1170005c7f9a21","last_reissued_at":"2026-05-18T00:55:34.513420Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:34.513420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $R$. 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