{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:4HI53HSILDQPYYT6NZCTNKDFTL","short_pith_number":"pith:4HI53HSI","canonical_record":{"source":{"id":"math/0609678","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.ST","submitted_at":"2006-09-25T03:19:54Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"80bab1ae6829cbbb00d3376288584183634813e8532241380bd821e0cf88a79f","abstract_canon_sha256":"4d9643f276c1235f442c8633aa22c93f848822c26d3a47626240adba8e14db24"},"schema_version":"1.0"},"canonical_sha256":"e1d1dd9e4858e0fc627e6e4536a8659ae9b6243bb8cd7c4e34825655d7944e37","source":{"kind":"arxiv","id":"math/0609678","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609678","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609678v2","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609678","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"pith_short_12","alias_value":"4HI53HSILDQP","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"4HI53HSILDQPYYT6","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"4HI53HSI","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:4HI53HSILDQPYYT6NZCTNKDFTL","target":"record","payload":{"canonical_record":{"source":{"id":"math/0609678","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.ST","submitted_at":"2006-09-25T03:19:54Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"80bab1ae6829cbbb00d3376288584183634813e8532241380bd821e0cf88a79f","abstract_canon_sha256":"4d9643f276c1235f442c8633aa22c93f848822c26d3a47626240adba8e14db24"},"schema_version":"1.0"},"canonical_sha256":"e1d1dd9e4858e0fc627e6e4536a8659ae9b6243bb8cd7c4e34825655d7944e37","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:53.236500Z","signature_b64":"gBelfb8z/ah4SD/v3iwG+RRWjM1uvKRLlrYfrYuLc9xOiktcEMnRAup2pNZ18HjRRQZIdyPHZsdUrUGCMS7DAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1d1dd9e4858e0fc627e6e4536a8659ae9b6243bb8cd7c4e34825655d7944e37","last_reissued_at":"2026-05-18T04:08:53.236031Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:53.236031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0609678","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:08:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U5hPpl1nXinssNcgw4AdfbT17ji4feqb6yGe+w8Onq8bRhHe6a9fMX2QWHD7Lxh+UJVjQW+rdUfyWsON6GL/DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T15:23:12.990806Z"},"content_sha256":"a99bc3d11598437a311936d1382134a18444f79bcbf776999999340c29070ddb","schema_version":"1.0","event_id":"sha256:a99bc3d11598437a311936d1382134a18444f79bcbf776999999340c29070ddb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:4HI53HSILDQPYYT6NZCTNKDFTL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Strong consistency of MLE for finite mixtures of location-scale distributions when the ratios of the scale parameters are exponentially small","license":"","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Kentaro Tanaka","submitted_at":"2006-09-25T03:19:54Z","abstract_excerpt":"In finite mixtures of location-scale distributions, if there is no constraint on the parameters then the maximum likelihood estimate does not exist. But when the ratios of the scale parameters are restricted appropriately, the maximum likelihood estimate exists. We prove that the maximum likelihood estimator (MLE) is strongly consistent, if the ratios of the scale parameters are restricted from below by $\\exp(-n^{d}), 0 < d < 1 $, where $n$ is the sample size."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609678","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:08:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IFLrsOUbBezg91hhQZ6kr7fJz5pYjMxisawpXMj3eS+QBP1FHeUzwsprqOXIqeVJqIi4i+JbX9i9X1I4QNgHDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T15:23:12.991459Z"},"content_sha256":"75f3788e702f3420c9a7869bf4f8d08a2e5a40b8e9974171b6692cbd20e41c84","schema_version":"1.0","event_id":"sha256:75f3788e702f3420c9a7869bf4f8d08a2e5a40b8e9974171b6692cbd20e41c84"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4HI53HSILDQPYYT6NZCTNKDFTL/bundle.json","state_url":"https://pith.science/pith/4HI53HSILDQPYYT6NZCTNKDFTL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4HI53HSILDQPYYT6NZCTNKDFTL/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":"2026-05-26T15:23:12Z","links":{"resolver":"https://pith.science/pith/4HI53HSILDQPYYT6NZCTNKDFTL","bundle":"https://pith.science/pith/4HI53HSILDQPYYT6NZCTNKDFTL/bundle.json","state":"https://pith.science/pith/4HI53HSILDQPYYT6NZCTNKDFTL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4HI53HSILDQPYYT6NZCTNKDFTL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:4HI53HSILDQPYYT6NZCTNKDFTL","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":"4d9643f276c1235f442c8633aa22c93f848822c26d3a47626240adba8e14db24","cross_cats_sorted":["stat.TH"],"license":"","primary_cat":"math.ST","submitted_at":"2006-09-25T03:19:54Z","title_canon_sha256":"80bab1ae6829cbbb00d3376288584183634813e8532241380bd821e0cf88a79f"},"schema_version":"1.0","source":{"id":"math/0609678","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609678","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609678v2","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609678","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"pith_short_12","alias_value":"4HI53HSILDQP","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"4HI53HSILDQPYYT6","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"4HI53HSI","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:75f3788e702f3420c9a7869bf4f8d08a2e5a40b8e9974171b6692cbd20e41c84","target":"graph","created_at":"2026-05-18T04:08:53Z","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":"In finite mixtures of location-scale distributions, if there is no constraint on the parameters then the maximum likelihood estimate does not exist. But when the ratios of the scale parameters are restricted appropriately, the maximum likelihood estimate exists. We prove that the maximum likelihood estimator (MLE) is strongly consistent, if the ratios of the scale parameters are restricted from below by $\\exp(-n^{d}), 0 < d < 1 $, where $n$ is the sample size.","authors_text":"Kentaro Tanaka","cross_cats":["stat.TH"],"headline":"","license":"","primary_cat":"math.ST","submitted_at":"2006-09-25T03:19:54Z","title":"Strong consistency of MLE for finite mixtures of location-scale distributions when the ratios of the scale parameters are exponentially small"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609678","kind":"arxiv","version":2},"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:a99bc3d11598437a311936d1382134a18444f79bcbf776999999340c29070ddb","target":"record","created_at":"2026-05-18T04:08:53Z","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":"4d9643f276c1235f442c8633aa22c93f848822c26d3a47626240adba8e14db24","cross_cats_sorted":["stat.TH"],"license":"","primary_cat":"math.ST","submitted_at":"2006-09-25T03:19:54Z","title_canon_sha256":"80bab1ae6829cbbb00d3376288584183634813e8532241380bd821e0cf88a79f"},"schema_version":"1.0","source":{"id":"math/0609678","kind":"arxiv","version":2}},"canonical_sha256":"e1d1dd9e4858e0fc627e6e4536a8659ae9b6243bb8cd7c4e34825655d7944e37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1d1dd9e4858e0fc627e6e4536a8659ae9b6243bb8cd7c4e34825655d7944e37","first_computed_at":"2026-05-18T04:08:53.236031Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:53.236031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gBelfb8z/ah4SD/v3iwG+RRWjM1uvKRLlrYfrYuLc9xOiktcEMnRAup2pNZ18HjRRQZIdyPHZsdUrUGCMS7DAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:53.236500Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0609678","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a99bc3d11598437a311936d1382134a18444f79bcbf776999999340c29070ddb","sha256:75f3788e702f3420c9a7869bf4f8d08a2e5a40b8e9974171b6692cbd20e41c84"],"state_sha256":"34d045701608b652367ab40a6fca56fe9557085c71c60be7c64c35e3298a51af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GiXNc/IG4pJHDxH/otkHDBoI+ygGElO9f420zem/jiO7NJ+dNiO1PIhiiUN2dd74wrDjAtPSUTSxOhenBfBtCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T15:23:12.993962Z","bundle_sha256":"e06fc79d6dde55b7de94a24de87bdfc717b8398ac175f1536022e2ca77e615fe"}}