{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6NIKTFKNQ4BB4RWPUS7TRTYLBN","short_pith_number":"pith:6NIKTFKN","canonical_record":{"source":{"id":"1509.02451","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2015-09-08T17:14:54Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"dd669a85033a5629fb7815f430eb2f16a2760f81626929701ca978c92f5e90dd","abstract_canon_sha256":"a651ef3b3967dbeac20f454bbe274cfc9d306dc2180a7eefb0ceb09eba13849c"},"schema_version":"1.0"},"canonical_sha256":"f350a9954d87021e46cfa4bf38cf0b0b76b86c7e8f453f1df089167e9f02ca07","source":{"kind":"arxiv","id":"1509.02451","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.02451","created_at":"2026-05-18T01:33:40Z"},{"alias_kind":"arxiv_version","alias_value":"1509.02451v1","created_at":"2026-05-18T01:33:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.02451","created_at":"2026-05-18T01:33:40Z"},{"alias_kind":"pith_short_12","alias_value":"6NIKTFKNQ4BB","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6NIKTFKNQ4BB4RWP","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6NIKTFKN","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6NIKTFKNQ4BB4RWPUS7TRTYLBN","target":"record","payload":{"canonical_record":{"source":{"id":"1509.02451","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2015-09-08T17:14:54Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"dd669a85033a5629fb7815f430eb2f16a2760f81626929701ca978c92f5e90dd","abstract_canon_sha256":"a651ef3b3967dbeac20f454bbe274cfc9d306dc2180a7eefb0ceb09eba13849c"},"schema_version":"1.0"},"canonical_sha256":"f350a9954d87021e46cfa4bf38cf0b0b76b86c7e8f453f1df089167e9f02ca07","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:40.860083Z","signature_b64":"VDw4NuNU0iiQk7yhC2z+YHuUtdaKUe5F/gVP4+cLtzx7K3ZAGxtlmbRqjSu7F6MgYePL9yQNdZgrfyKJ88OrAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f350a9954d87021e46cfa4bf38cf0b0b76b86c7e8f453f1df089167e9f02ca07","last_reissued_at":"2026-05-18T01:33:40.859543Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:40.859543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.02451","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:33:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OatEin4/QnxaTPIX80mKxLHPIMadi0xaJMepPKL6/y/IOIeguPA8GXgqE91f9VfbHCKKPpOkXe0lhsQcqLCUBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T20:00:56.139245Z"},"content_sha256":"2df8cbf258937b9de26fc7bde3d5f06493d6909467ed9321631feb2c6ed67f40","schema_version":"1.0","event_id":"sha256:2df8cbf258937b9de26fc7bde3d5f06493d6909467ed9321631feb2c6ed67f40"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6NIKTFKNQ4BB4RWPUS7TRTYLBN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Improved Second Order Estimation in the Singular Multivariate Normal Model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Didier Ch\\'etelat, Martin T. Wells","submitted_at":"2015-09-08T17:14:54Z","abstract_excerpt":"We consider the problem of estimating covariance and precision matrices, and their associated discriminant coefficients, from normal data when the rank of the covariance matrix is strictly smaller than its dimension and the available sample size. Using unbiased risk estimation, we construct novel estimators by minimizing upper bounds on the difference in risk over several classes. Our proposal estimates are empirically demonstrated to offer substantial improvement over classical approaches."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02451","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:33:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EuYfuHTyGLsPwTpeKy87Bh0lvCumYCPClDkh1u0qsfGrg5g91KP99c2eSiksfcQ1ZcW2n9YoCAe91RcqqsNvBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T20:00:56.139817Z"},"content_sha256":"9e2a947cfb7a7f951dc073783543f662c45916b674c409da434b9efd95caac59","schema_version":"1.0","event_id":"sha256:9e2a947cfb7a7f951dc073783543f662c45916b674c409da434b9efd95caac59"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6NIKTFKNQ4BB4RWPUS7TRTYLBN/bundle.json","state_url":"https://pith.science/pith/6NIKTFKNQ4BB4RWPUS7TRTYLBN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6NIKTFKNQ4BB4RWPUS7TRTYLBN/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-19T20:00:56Z","links":{"resolver":"https://pith.science/pith/6NIKTFKNQ4BB4RWPUS7TRTYLBN","bundle":"https://pith.science/pith/6NIKTFKNQ4BB4RWPUS7TRTYLBN/bundle.json","state":"https://pith.science/pith/6NIKTFKNQ4BB4RWPUS7TRTYLBN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6NIKTFKNQ4BB4RWPUS7TRTYLBN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6NIKTFKNQ4BB4RWPUS7TRTYLBN","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":"a651ef3b3967dbeac20f454bbe274cfc9d306dc2180a7eefb0ceb09eba13849c","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2015-09-08T17:14:54Z","title_canon_sha256":"dd669a85033a5629fb7815f430eb2f16a2760f81626929701ca978c92f5e90dd"},"schema_version":"1.0","source":{"id":"1509.02451","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.02451","created_at":"2026-05-18T01:33:40Z"},{"alias_kind":"arxiv_version","alias_value":"1509.02451v1","created_at":"2026-05-18T01:33:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.02451","created_at":"2026-05-18T01:33:40Z"},{"alias_kind":"pith_short_12","alias_value":"6NIKTFKNQ4BB","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6NIKTFKNQ4BB4RWP","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6NIKTFKN","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:9e2a947cfb7a7f951dc073783543f662c45916b674c409da434b9efd95caac59","target":"graph","created_at":"2026-05-18T01:33:40Z","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":"We consider the problem of estimating covariance and precision matrices, and their associated discriminant coefficients, from normal data when the rank of the covariance matrix is strictly smaller than its dimension and the available sample size. Using unbiased risk estimation, we construct novel estimators by minimizing upper bounds on the difference in risk over several classes. Our proposal estimates are empirically demonstrated to offer substantial improvement over classical approaches.","authors_text":"Didier Ch\\'etelat, Martin T. Wells","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2015-09-08T17:14:54Z","title":"Improved Second Order Estimation in the Singular Multivariate Normal Model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02451","kind":"arxiv","version":1},"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":"sha256:2df8cbf258937b9de26fc7bde3d5f06493d6909467ed9321631feb2c6ed67f40","target":"record","created_at":"2026-05-18T01:33:40Z","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":"a651ef3b3967dbeac20f454bbe274cfc9d306dc2180a7eefb0ceb09eba13849c","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2015-09-08T17:14:54Z","title_canon_sha256":"dd669a85033a5629fb7815f430eb2f16a2760f81626929701ca978c92f5e90dd"},"schema_version":"1.0","source":{"id":"1509.02451","kind":"arxiv","version":1}},"canonical_sha256":"f350a9954d87021e46cfa4bf38cf0b0b76b86c7e8f453f1df089167e9f02ca07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f350a9954d87021e46cfa4bf38cf0b0b76b86c7e8f453f1df089167e9f02ca07","first_computed_at":"2026-05-18T01:33:40.859543Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:40.859543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VDw4NuNU0iiQk7yhC2z+YHuUtdaKUe5F/gVP4+cLtzx7K3ZAGxtlmbRqjSu7F6MgYePL9yQNdZgrfyKJ88OrAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:40.860083Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.02451","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2df8cbf258937b9de26fc7bde3d5f06493d6909467ed9321631feb2c6ed67f40","sha256:9e2a947cfb7a7f951dc073783543f662c45916b674c409da434b9efd95caac59"],"state_sha256":"ffe3e9e2dd9dfa8b8ba8df8659f1cc3c89c2c03638d4e93b2c25058fb8e777b2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4UXIkJ8iyEFFfFRU5s2W13rVFfirFKzZA4ijPcQRQC/IW8MO7J7n7U/ZuVpSRHNPAkkoC+iS6H0eSOkIGNFVAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-19T20:00:56.143337Z","bundle_sha256":"0e1ffc5b6d00c0961d4940e37058725c747d1e8f5a769f90847317fb2fa9dc87"}}