DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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Sufficient statistics

Definition

Let $((X, A), Θ, (P_{θ}))$ be a statistical experiment, where $(X, A)$ is standard Borel, and let $T : (X, A) \to (Y, B)$ be a statistic with standard Borel target. Write $P_{θ}^{T} = P_{θ} \circ T^{- 1}$ . The statistic $T$ is sufficient if there is one probability kernel $K$ from $(Y, B)$ to $(X, A)$ such that, for every $θ \in Θ$ , $A \in A$ , and $C \in B$ ,

$P_{θ} (A \cap T^{- 1} (C)) = \int_{C} K (y, A) P_{θ}^{T} (d y) .$

Thus $K (T (x), A)$ is a common version of the conditional probability of $A$ given $T$ , under every parameter. The kernel is fixed independently of $θ$ . Under AC, standard Borelness gives each law an individual conditional law through Existence of regular conditional distributions for standard borel targets and Regular conditional kernels factor through a standard borel conditioning variable; sufficiency additionally asks that one kernel works across parameters.

Depends on

Used by

An external dominating law need not be a valid sufficiency reference Counterexample
An identifiable model can have persistent ABC summary distortion Counterexample
Minimal sufficient sigma-fields Definition
Minimal sufficient statistics Definition
A Bernoulli count is a sufficient summary for a finite-sample posterior Example
Rao–Blackwellizing one Bernoulli observation by the sample sum Example
The Bernoulli count is minimal sufficient Example
A countable mixture likelihood-ratio vector is minimal sufficient Theorem
A sufficient summary preserves the Bayesian posterior Theorem
Basu’s independence theorem Theorem
Common-reference factorization criterion for sufficiency Theorem
Countable factorization criterion for sufficiency Theorem
Lehmann–Scheffé theorem Theorem
Likelihood-ratio classes are minimal sufficient on a common discrete support Theorem
Rao–Blackwell theorem for scalar finite-variance estimators Theorem
Standard Borel dominated factorization criterion Theorem
Sufficiency makes the conditional rejection probability nuisance-free Theorem
The canonical statistic of an exponential family is sufficient Theorem
The sum of exponential-family statistics is sufficient for an IID sample Theorem

Dependency tree · two levels

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Sources

Ryan Martin, Lecture Notes on Advanced Statistical Theory (standard reference, not scraped)
Joshua M. Tebbs, STAT 512 Mathematical Statistics Lecture Notes (standard reference, not scraped)
Peter Kempthorne, MIT 18.655 Lecture Note 6: Sufficiency (standard reference, not scraped)
Noah Fithian, Stat 210a Reader: Sufficiency (standard reference, not scraped)
Noah Fithian, Stat 210a Reader: Completeness, Ancillarity, and Basu’s Theorem (standard reference, not scraped)
Charles J. Geyer, Stat 5421 Lecture Notes: Exponential Families (standard reference, not scraped)
Jiahao Tian, Chapter 2: Sufficiency and Exponential Families (standard reference, not scraped)