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The canonical statistic of an exponential family is sufficient

Statement

Assume AC. Let $(P_{η})_{η \in Ω}$ be the exponential family of Exponential families and full rank on a standard Borel sample space, with $Ω \neq \emptyset$ , and let $T$ be its canonical statistic. Then $T$ is sufficient.

Facts & Assumptions

Given: AC, the measurable carrier $h$ , sigma-finite base $μ$ , nonempty natural parameter set $Ω$ , log normalizer $A$ , and density formula from the exponential-family definition.

[A1]

The Axiom of Choice: the common-reference RN criterion is used under AC.

[F1]

Exponential families and full rank: $p_{η} = h \exp (η \cdot T - A (η))$ and $A = \log \int h e^{η \cdot T} d μ$ .

[F2]

Sufficient statistics: a common regular conditional law of the data given $T$ is the sufficiency criterion.

[F3]

Common-reference factorization criterion for sufficiency: with a reference probability that is a member of the family, $T$ is sufficient when every RN derivative relative to it is a Borel function of $T$ .

Proof

1.1F1givenalgebra

Fix any $η_{0} \in Ω$ ; nonemptiness is part of the family being considered. The density in [F1] is positive exactly on ${h > 0}$ for every parameter. For $η \in Ω$ , dividing the densities there gives $d P_{η} / d P_{η_{0}} = \exp {(η - η_{0}) \cdot T - (A (η) - A (η_{0}))}$ $P_{η_{0}}$ -almost surely. The right side is a Borel function of $T$ .

2.1A1F2F3step 1.1given∎

The reference $P_{η_{0}}$ is a member of the model and dominates every other member because the densities have the common support ${h > 0}$ . Apply [F3] to the derivative in step 1.1; its criterion yields a common conditional law of $X$ given $T$ . By [F2], $T$ is sufficient.

Depends on

Used by

Normal sample mean and residuals are independent Example

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)