{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:FOUHQ4KM5Z2UAHLYI7D7AGWNXM","short_pith_number":"pith:FOUHQ4KM","schema_version":"1.0","canonical_sha256":"2ba878714cee75401d7847c7f01acdbb2602de8c8e49fcebc1dd85d0181fb444","source":{"kind":"arxiv","id":"1205.4219","version":2},"attestation_state":"computed","paper":{"title":"Optimal hypothesis testing for high dimensional covariance matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"T. Tony Cai, Zongming Ma","submitted_at":"2012-05-18T19:06:15Z","abstract_excerpt":"This paper considers testing a covariance matrix $\\Sigma$ in the high dimensional setting where the dimension $p$ can be comparable or much larger than the sample size $n$. The problem of testing the hypothesis $H_0:\\Sigma=\\Sigma_0$ for a given covariance matrix $\\Sigma_0$ is studied from a minimax point of view. We first characterize the boundary that separates the testable region from the non-testable region by the Frobenius norm when the ratio between the dimension $p$ over the sample size $n$ is bounded. A test based on a $U$-statistic is introduced and is shown to be rate optimal over thi"},"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":"1205.4219","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2012-05-18T19:06:15Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"03b42b14ca98a1e01bb805d8923a59e1f47efe6358c02eea1f9d0ece10b2a4bd","abstract_canon_sha256":"6ac7f23c148f27382a5ba8d3fb9fe283bcc3c4586ac4453681336fea24fec2ac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:24.366485Z","signature_b64":"xEkL2Zvb+brJA/R6ykbb1TEiVC8MEOBIKogmK3wcL17Vf6sWn+PVCX6O8YQaBgtk/4jVfwSLaGVBj9A8DNg+AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ba878714cee75401d7847c7f01acdbb2602de8c8e49fcebc1dd85d0181fb444","last_reissued_at":"2026-05-18T03:04:24.365943Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:24.365943Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal hypothesis testing for high dimensional covariance matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"T. Tony Cai, Zongming Ma","submitted_at":"2012-05-18T19:06:15Z","abstract_excerpt":"This paper considers testing a covariance matrix $\\Sigma$ in the high dimensional setting where the dimension $p$ can be comparable or much larger than the sample size $n$. The problem of testing the hypothesis $H_0:\\Sigma=\\Sigma_0$ for a given covariance matrix $\\Sigma_0$ is studied from a minimax point of view. We first characterize the boundary that separates the testable region from the non-testable region by the Frobenius norm when the ratio between the dimension $p$ over the sample size $n$ is bounded. A test based on a $U$-statistic is introduced and is shown to be rate optimal over thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4219","kind":"arxiv","version":2},"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":"1205.4219","created_at":"2026-05-18T03:04:24.366022+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.4219v2","created_at":"2026-05-18T03:04:24.366022+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4219","created_at":"2026-05-18T03:04:24.366022+00:00"},{"alias_kind":"pith_short_12","alias_value":"FOUHQ4KM5Z2U","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_16","alias_value":"FOUHQ4KM5Z2UAHLY","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_8","alias_value":"FOUHQ4KM","created_at":"2026-05-18T12:27:06.952714+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/FOUHQ4KM5Z2UAHLYI7D7AGWNXM","json":"https://pith.science/pith/FOUHQ4KM5Z2UAHLYI7D7AGWNXM.json","graph_json":"https://pith.science/api/pith-number/FOUHQ4KM5Z2UAHLYI7D7AGWNXM/graph.json","events_json":"https://pith.science/api/pith-number/FOUHQ4KM5Z2UAHLYI7D7AGWNXM/events.json","paper":"https://pith.science/paper/FOUHQ4KM"},"agent_actions":{"view_html":"https://pith.science/pith/FOUHQ4KM5Z2UAHLYI7D7AGWNXM","download_json":"https://pith.science/pith/FOUHQ4KM5Z2UAHLYI7D7AGWNXM.json","view_paper":"https://pith.science/paper/FOUHQ4KM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.4219&json=true","fetch_graph":"https://pith.science/api/pith-number/FOUHQ4KM5Z2UAHLYI7D7AGWNXM/graph.json","fetch_events":"https://pith.science/api/pith-number/FOUHQ4KM5Z2UAHLYI7D7AGWNXM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FOUHQ4KM5Z2UAHLYI7D7AGWNXM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FOUHQ4KM5Z2UAHLYI7D7AGWNXM/action/storage_attestation","attest_author":"https://pith.science/pith/FOUHQ4KM5Z2UAHLYI7D7AGWNXM/action/author_attestation","sign_citation":"https://pith.science/pith/FOUHQ4KM5Z2UAHLYI7D7AGWNXM/action/citation_signature","submit_replication":"https://pith.science/pith/FOUHQ4KM5Z2UAHLYI7D7AGWNXM/action/replication_record"}},"created_at":"2026-05-18T03:04:24.366022+00:00","updated_at":"2026-05-18T03:04:24.366022+00:00"}