{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:KPDZJ7RP46PX3FNGFCEO6RDMIJ","short_pith_number":"pith:KPDZJ7RP","schema_version":"1.0","canonical_sha256":"53c794fe2fe79f7d95a62888ef446c4241e611b87f925443b287167c3df576f4","source":{"kind":"arxiv","id":"1111.5934","version":3},"attestation_state":"computed","paper":{"title":"The limit distribution of the $L_{\\infty}$-error of Grenander-type estimators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"C\\'ecile Durot, Hendrik P. Lopuha\\\"a, Vladimir N. Kulikov","submitted_at":"2011-11-25T10:13:17Z","abstract_excerpt":"Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n/\\log n)^{-1/3}$ and the limiting distribution to be Gumbel."},"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":"1111.5934","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2011-11-25T10:13:17Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"b7d8879877ab526a8501625858488e8a488f848c5e660130bbacc510bd0b02bb","abstract_canon_sha256":"e9bf87db0523b2e8dc2663a490be78121d43bc50a0b1c318de0f17d35e9d309c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:49.353520Z","signature_b64":"W/9S+In6u7yFYjtSQrF3m4bC/vAmYlNeobwyh5+4qLh7/7DPU1OiiUqTMvhCkkJlFKbBvuU1hLf5ESKP81fHDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53c794fe2fe79f7d95a62888ef446c4241e611b87f925443b287167c3df576f4","last_reissued_at":"2026-05-18T03:44:49.352793Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:49.352793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The limit distribution of the $L_{\\infty}$-error of Grenander-type estimators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"C\\'ecile Durot, Hendrik P. Lopuha\\\"a, Vladimir N. Kulikov","submitted_at":"2011-11-25T10:13:17Z","abstract_excerpt":"Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n/\\log n)^{-1/3}$ and the limiting distribution to be Gumbel."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5934","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1111.5934","created_at":"2026-05-18T03:44:49.352903+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.5934v3","created_at":"2026-05-18T03:44:49.352903+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5934","created_at":"2026-05-18T03:44:49.352903+00:00"},{"alias_kind":"pith_short_12","alias_value":"KPDZJ7RP46PX","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_16","alias_value":"KPDZJ7RP46PX3FNG","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_8","alias_value":"KPDZJ7RP","created_at":"2026-05-18T12:26:32.869790+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KPDZJ7RP46PX3FNGFCEO6RDMIJ","json":"https://pith.science/pith/KPDZJ7RP46PX3FNGFCEO6RDMIJ.json","graph_json":"https://pith.science/api/pith-number/KPDZJ7RP46PX3FNGFCEO6RDMIJ/graph.json","events_json":"https://pith.science/api/pith-number/KPDZJ7RP46PX3FNGFCEO6RDMIJ/events.json","paper":"https://pith.science/paper/KPDZJ7RP"},"agent_actions":{"view_html":"https://pith.science/pith/KPDZJ7RP46PX3FNGFCEO6RDMIJ","download_json":"https://pith.science/pith/KPDZJ7RP46PX3FNGFCEO6RDMIJ.json","view_paper":"https://pith.science/paper/KPDZJ7RP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.5934&json=true","fetch_graph":"https://pith.science/api/pith-number/KPDZJ7RP46PX3FNGFCEO6RDMIJ/graph.json","fetch_events":"https://pith.science/api/pith-number/KPDZJ7RP46PX3FNGFCEO6RDMIJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KPDZJ7RP46PX3FNGFCEO6RDMIJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KPDZJ7RP46PX3FNGFCEO6RDMIJ/action/storage_attestation","attest_author":"https://pith.science/pith/KPDZJ7RP46PX3FNGFCEO6RDMIJ/action/author_attestation","sign_citation":"https://pith.science/pith/KPDZJ7RP46PX3FNGFCEO6RDMIJ/action/citation_signature","submit_replication":"https://pith.science/pith/KPDZJ7RP46PX3FNGFCEO6RDMIJ/action/replication_record"}},"created_at":"2026-05-18T03:44:49.352903+00:00","updated_at":"2026-05-18T03:44:49.352903+00:00"}