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    "title": "Basu: when an ancillary statistic is independent of a sufficient summary",
    "summary": "Basu's1955 Theorem2 shows that a boundedly complete sufficient statistic is independent of any statistic whose distribution does not depend on the model parameter.\n\nA statistic may summarize everything in a model about its unknown parameter, while another has the same distribution for every parameter value. These properties alone need not make the two independent. Basu identifies bounded completeness as the extra condition, gives a short conditional-probability proof and uses it to recover specified normal and Gamma distribution independence results.",
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        "description": "The exact target is Theorem 2 under the measurable-family and conditional-probability/sufficiency definitions of original 377–379, together with its stated illustrative use. Bounded completeness is required; arbitrary sufficient statistics do not yield the converse, as the original explicitly warns. Independence holds for each member of the model family, with the source's almost-everywhere conventions; no simultaneous pointwise strengthening across every conditional event/parameter is asserted. Parameter-independent distribution is the meaning of ancillary here, not absence of all scientific information. The normal and Gamma examples retain their distribution/model assumptions. The paper's Gaussian characterization draws on Darmois's earlier result and its Gamma characterization invokes Laha; neither is newly attributed wholesale to this criterion. No independent proof reconstruction or formal verification is recorded. Later generalizations and every statistical use of completeness remain separate.",
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      "statement": "# Basu: when an ancillary statistic is independent of a sufficient summary\n\nBasu's1955 Theorem2 shows that a boundedly complete sufficient statistic is independent of any statistic whose distribution does not depend on the model parameter.\n\nA statistic may summarize everything in a model about its unknown parameter, while another has the same distribution for every parameter value. These properties alone need not make the two independent. Basu identifies bounded completeness as the extra condition, gives a short conditional-probability proof and uses it to recover specified normal and Gamma distribution independence results.\n\n## Scientific scope\n\nThe exact target is Theorem 2 under the measurable-family and conditional-probability/sufficiency definitions of original 377–379, together with its stated illustrative use. Bounded completeness is required; arbitrary sufficient statistics do not yield the converse, as the original explicitly warns. Independence holds for each member of the model family, with the source's almost-everywhere conventions; no simultaneous pointwise strengthening across every conditional event/parameter is asserted. Parameter-independent distribution is the meaning of ancillary here, not absence of all scientific information. The normal and Gamma examples retain their distribution/model assumptions. The paper's Gaussian characterization draws on Darmois's earlier result and its Gamma characterization invokes Laha; neither is newly attributed wholesale to this criterion. No independent proof reconstruction or formal verification is recorded. Later generalizations and every statistical use of completeness remain separate.\n\n## People and systems\n\nDebabrata Basu\n\n\n\n## Original sources\n\n- [D.Basu, On Statistics Independent of a Complete Sufficient Statistic, Sankhya15(1955),377–380; facsimile in2011 Selected Works](https://link.springer.com/content/pdf/10.1007/978-1-4419-5825-9_14.pdf)\n- [University of Washington statistics reading note: Basu1955 original citation](https://sites.stat.washington.edu/peter/582/basu.html)\n",
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