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We consider the problem of estimating the $r$-order missing mass, which is a discrete functional of $P$ defined as $$\\theta_{r}(P;\\mathbf{X}_{n})=\\sum_{j\\geq1}p^{r}_{j}I(Y_{n,j}=0).$$ This is generalization of the missing mass whose estimation is a classical problem in statistics, being the subject of numerous studies both in theory a"},"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":"2306.14998","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2023-06-26T18:29:07Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"342147686c430e648aa1f216d522af43474a55eee139d293f6c404a026629d62","abstract_canon_sha256":"9136f7a4084afddcfc208adb62acd4778d31cdc6d7468484e5dfe0c9be39bf25"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:42:29.565951Z","signature_b64":"yhUb10h+5ENVo/hFWu/r1Aitle3zd4gCmxmUAoVOCWpsn3S2nuxSZmglJOro/A7HlhcvxaBJdJ0JRcWspfNTCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a9ffffa867bf7a4bf4c089d22c379bc119ca5fb680b7f7bb644b0fd6e2d9ce0","last_reissued_at":"2026-07-05T08:42:29.565490Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:42:29.565490Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal estimation of high-order missing masses, and the rare-type match problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Stefano Favaro, Zacharie Naulet","submitted_at":"2023-06-26T18:29:07Z","abstract_excerpt":"Consider a random sample $(X_{1},\\ldots,X_{n})$ from an unknown discrete distribution $P=\\sum_{j\\geq1}p_{j}\\delta_{s_{j}}$ on a countable alphabet $\\mathbb{S}$, and let $(Y_{n,j})_{j\\geq1}$ be the empirical frequencies of distinct symbols $s_{j}$'s in the sample. 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