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Inverting the tests for observations in $[0,1]$ gives the confidence interval studied by Learned-Miller and Thomas (2020), which reduces to Clopper--Pearson for Bernoulli data.\n  We give a finite- and large-sample account of Gaffke's te"},"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":"2607.18661","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-21T03:14:21Z","cross_cats_sorted":["cs.IT","eess.SP","math.IT","math.PR","stat.ME","stat.TH"],"title_canon_sha256":"d4672f780c3d204358d4afecb2a0fb4cd576258d07716b90ef9d42cbbba8f61d","abstract_canon_sha256":"9a500ee1356d6ca4bd3f5a1977e36e08ba164fe3a6099ec1934502acdaeeb2df"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-22T00:22:59.159245Z","signature_b64":"ZL+4qSoDEzhOqnaUUIz9bsy3x43sLpgS8t9s9SSMEJncXrLVPyn70GZ6z+P2ojF64BbPjAnqdMxbbHnI2mluDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92ffa2ff1f387c4a358e28ef77a3bfaf30ca35be614d51e009beed5ebe8e6e26","last_reissued_at":"2026-07-22T00:22:59.158359Z","signature_status":"signed_v1","first_computed_at":"2026-07-22T00:22:59.158359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gaffke's confidence interval for the mean of bounded data is inadmissible but asymptotically efficient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","eess.SP","math.IT","math.PR","stat.ME","stat.TH"],"primary_cat":"math.ST","authors_text":"Aaditya Ramdas, Ian Waudby-Smith, Jiahao Ming, Ruodu Wang, Yi Shen","submitted_at":"2026-07-21T03:14:21Z","abstract_excerpt":"Given observations $\\mathbf x=(x_1,\\dots,x_n)$, Gaffke (2005) defined \\[ K_n(\\mathbf x)=\\mathbb{P}_{\\mathbf D}\\!\\left\\{\\sum_{i=1}^n x_iD_i\\le 1\\right\\}, \\qquad (D_0,D_1,\\ldots,D_n)\\sim\\mathrm{Dirichlet}(1,\\ldots,1), \\] and conjectured that it is a $p$-value whenever the inputs are independent e-values. 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