{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:Y6R7CFQNB2ZJQRQ6XGNYWLPFWL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c9f8b9eb918064fa427efa066db59e6318bb620fb61ba8dd242f5cc1e7fa8d0a","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2014-11-19T18:28:32Z","title_canon_sha256":"b3225d73ad3a50611aac855edbb7cefe7319b9197b0b9999b58af43d54a0fa4f"},"schema_version":"1.0","source":{"id":"1411.5305","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.5305","created_at":"2026-05-18T02:33:24Z"},{"alias_kind":"arxiv_version","alias_value":"1411.5305v2","created_at":"2026-05-18T02:33:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.5305","created_at":"2026-05-18T02:33:24Z"},{"alias_kind":"pith_short_12","alias_value":"Y6R7CFQNB2ZJ","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"Y6R7CFQNB2ZJQRQ6","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"Y6R7CFQN","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:b57ba46ad80b2e73f4c127295a18061aa02315233ebf3ef68bdec09441b807c1","target":"graph","created_at":"2026-05-18T02:33:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Assume that X is a set of sample statistics which follow a special case Central Limit Theorem, namely: as the sample size n increases the corresponding distribution becomes multivariate Normal with the mean (of each X) equal to zero and with an idempotent variance-covariance matrix V. It is well known that X^{T}X has (in the same limit), a chi-squared distribution with degrees of freedom equal to the trace of V. In this article we extend the above result to include the corresponding (1/n)-proportional corrections, making the new approximation substantially more accurate and extending its range","authors_text":"Hao Yuan Zhang, Jan Vrbik","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2014-11-19T18:28:32Z","title":"Accurate distribution of X^{T}X with singular, idempotent variance-covariance matrix"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.5305","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:56fab224124fed2b60ace9432fd9f26ce8fc27dba46d97e6ee33acb46e3e9e04","target":"record","created_at":"2026-05-18T02:33:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c9f8b9eb918064fa427efa066db59e6318bb620fb61ba8dd242f5cc1e7fa8d0a","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2014-11-19T18:28:32Z","title_canon_sha256":"b3225d73ad3a50611aac855edbb7cefe7319b9197b0b9999b58af43d54a0fa4f"},"schema_version":"1.0","source":{"id":"1411.5305","kind":"arxiv","version":2}},"canonical_sha256":"c7a3f1160d0eb298461eb99b8b2de5b2c50f3a981c363e276e86de5731f675b7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c7a3f1160d0eb298461eb99b8b2de5b2c50f3a981c363e276e86de5731f675b7","first_computed_at":"2026-05-18T02:33:24.842191Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:33:24.842191Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WBpkf5FtrfqqD4wiktzN8lOO3r/nJ3pf5gpAXcJH/Fz6Juh8HFxb3ERfPdEEJDTcLpVQU19ZGyLlJAbYNW27Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:33:24.842818Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.5305","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:56fab224124fed2b60ace9432fd9f26ce8fc27dba46d97e6ee33acb46e3e9e04","sha256:b57ba46ad80b2e73f4c127295a18061aa02315233ebf3ef68bdec09441b807c1"],"state_sha256":"d87ad2bfa0f4d59bfae0d8591098a9727baf8f4d94f86ebc26f8e98f9f744057"}