{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:6VOOTM5ALZD7352O77WEQ7SHWE","short_pith_number":"pith:6VOOTM5A","schema_version":"1.0","canonical_sha256":"f55ce9b3a05e47fdf74effec487e47b102e5ffdf3344ca0d78edb4f2190cc152","source":{"kind":"arxiv","id":"math/0602244","version":1},"attestation_state":"computed","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"},"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":"math/0602244","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.ST","submitted_at":"2006-02-11T13:40:14Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"f46369627bb1d4ee860e5bf2d8a1c96fd9100b4d6cca795db0eda6255964164e","abstract_canon_sha256":"10e7568f4c203899a49208e081164c8df9d5622314fd330702d5724ab7320592"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:50.562215Z","signature_b64":"gKl7T+BBpWvixCT9TqsxMrDuBwQUtmg1BwDfHUtXnwxjDSvPhJEiC+xu4BrH8Hjg8svzC17PJ+YinXNlbRtyDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f55ce9b3a05e47fdf74effec487e47b102e5ffdf3344ca0d78edb4f2190cc152","last_reissued_at":"2026-05-18T01:08:50.561713Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:50.561713Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0602244","created_at":"2026-05-18T01:08:50.561792+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0602244v1","created_at":"2026-05-18T01:08:50.561792+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0602244","created_at":"2026-05-18T01:08:50.561792+00:00"},{"alias_kind":"pith_short_12","alias_value":"6VOOTM5ALZD7","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"6VOOTM5ALZD7352O","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"6VOOTM5A","created_at":"2026-05-18T12:25:53.939244+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/6VOOTM5ALZD7352O77WEQ7SHWE","json":"https://pith.science/pith/6VOOTM5ALZD7352O77WEQ7SHWE.json","graph_json":"https://pith.science/api/pith-number/6VOOTM5ALZD7352O77WEQ7SHWE/graph.json","events_json":"https://pith.science/api/pith-number/6VOOTM5ALZD7352O77WEQ7SHWE/events.json","paper":"https://pith.science/paper/6VOOTM5A"},"agent_actions":{"view_html":"https://pith.science/pith/6VOOTM5ALZD7352O77WEQ7SHWE","download_json":"https://pith.science/pith/6VOOTM5ALZD7352O77WEQ7SHWE.json","view_paper":"https://pith.science/paper/6VOOTM5A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0602244&json=true","fetch_graph":"https://pith.science/api/pith-number/6VOOTM5ALZD7352O77WEQ7SHWE/graph.json","fetch_events":"https://pith.science/api/pith-number/6VOOTM5ALZD7352O77WEQ7SHWE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6VOOTM5ALZD7352O77WEQ7SHWE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6VOOTM5ALZD7352O77WEQ7SHWE/action/storage_attestation","attest_author":"https://pith.science/pith/6VOOTM5ALZD7352O77WEQ7SHWE/action/author_attestation","sign_citation":"https://pith.science/pith/6VOOTM5ALZD7352O77WEQ7SHWE/action/citation_signature","submit_replication":"https://pith.science/pith/6VOOTM5ALZD7352O77WEQ7SHWE/action/replication_record"}},"created_at":"2026-05-18T01:08:50.561792+00:00","updated_at":"2026-05-18T01:08:50.561792+00:00"}