{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:UUZR3VE4TEEED5JRNP2D5VKOHH","short_pith_number":"pith:UUZR3VE4","schema_version":"1.0","canonical_sha256":"a5331dd49c990841f5316bf43ed54e39e026c72b4ae5211f28d7a3efac372378","source":{"kind":"arxiv","id":"1807.11260","version":1},"attestation_state":"computed","paper":{"title":"Cumulative distribution functions for the five simplest natural exponential families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"G\\'erard Letac","submitted_at":"2018-07-30T09:55:47Z","abstract_excerpt":"Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\\E(z^{X_a})=f(az)/f(a)$. Assume that $\\Pr(X_a\\leq k)$ has the form $c_k\\int_a ^{\\infty}u^k\\mu(du)$ for some number $c_k$ and some positive measure $\\mu,$ both independent of $a.$ We show that this asumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamm"},"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":"1807.11260","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-07-30T09:55:47Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"302d411cf563cb3056a58d68fea10c3ec11f4378c0a254b1448a0d4ac0a758d8","abstract_canon_sha256":"069f6f05e94b7b64981b499db12991689d5e255f4bd5c14fd3202084d0e51f2c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:32.475584Z","signature_b64":"U/qQKxpvhEUA0KPE+Y5mXZtDFcGICKMLRNi9QrSVjcFjeBUAEFVF/816yZeTjWof02JQasiie+3WV6MevXV8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5331dd49c990841f5316bf43ed54e39e026c72b4ae5211f28d7a3efac372378","last_reissued_at":"2026-05-18T00:09:32.474928Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:32.474928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cumulative distribution functions for the five simplest natural exponential families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"G\\'erard Letac","submitted_at":"2018-07-30T09:55:47Z","abstract_excerpt":"Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\\E(z^{X_a})=f(az)/f(a)$. Assume that $\\Pr(X_a\\leq k)$ has the form $c_k\\int_a ^{\\infty}u^k\\mu(du)$ for some number $c_k$ and some positive measure $\\mu,$ both independent of $a.$ We show that this asumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.11260","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":"1807.11260","created_at":"2026-05-18T00:09:32.475016+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.11260v1","created_at":"2026-05-18T00:09:32.475016+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.11260","created_at":"2026-05-18T00:09:32.475016+00:00"},{"alias_kind":"pith_short_12","alias_value":"UUZR3VE4TEEE","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"UUZR3VE4TEEED5JR","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"UUZR3VE4","created_at":"2026-05-18T12:32:56.356000+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/UUZR3VE4TEEED5JRNP2D5VKOHH","json":"https://pith.science/pith/UUZR3VE4TEEED5JRNP2D5VKOHH.json","graph_json":"https://pith.science/api/pith-number/UUZR3VE4TEEED5JRNP2D5VKOHH/graph.json","events_json":"https://pith.science/api/pith-number/UUZR3VE4TEEED5JRNP2D5VKOHH/events.json","paper":"https://pith.science/paper/UUZR3VE4"},"agent_actions":{"view_html":"https://pith.science/pith/UUZR3VE4TEEED5JRNP2D5VKOHH","download_json":"https://pith.science/pith/UUZR3VE4TEEED5JRNP2D5VKOHH.json","view_paper":"https://pith.science/paper/UUZR3VE4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.11260&json=true","fetch_graph":"https://pith.science/api/pith-number/UUZR3VE4TEEED5JRNP2D5VKOHH/graph.json","fetch_events":"https://pith.science/api/pith-number/UUZR3VE4TEEED5JRNP2D5VKOHH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UUZR3VE4TEEED5JRNP2D5VKOHH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UUZR3VE4TEEED5JRNP2D5VKOHH/action/storage_attestation","attest_author":"https://pith.science/pith/UUZR3VE4TEEED5JRNP2D5VKOHH/action/author_attestation","sign_citation":"https://pith.science/pith/UUZR3VE4TEEED5JRNP2D5VKOHH/action/citation_signature","submit_replication":"https://pith.science/pith/UUZR3VE4TEEED5JRNP2D5VKOHH/action/replication_record"}},"created_at":"2026-05-18T00:09:32.475016+00:00","updated_at":"2026-05-18T00:09:32.475016+00:00"}