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The Bernoulli count is minimal sufficient

Example

Assume AC. Let $X_{1}, \dots, X_{n}$ be iid Bernoulli $(p)$ with $p \in (0, 1)$ , sample space ${0, 1}^{n}$ , and $T (x) = \sum_{i} x_{i}$ . Then $T$ is minimal sufficient.

Facts & Assumptions

Given: AC, $n \geq 1$ , $p \in (0, 1)$ , the Bernoulli product family, and counting measure as the common reference, so its likelihood densities equal the point masses.

[A1]

The Axiom of Choice: AC is the axiom branch of the discrete minimal-sufficiency theorem.

[F1]

Likelihood-ratio equivalence on common discrete support: points are equivalent exactly when their point-mass ratio is constant in the parameter; with counting measure these are also the likelihood-density ratios.

[F2]

Likelihood-ratio classes are minimal sufficient on a common discrete support: the equivalence-class statistic is minimal sufficient on a countable common positive support.

Proof

1.1F1givenalgebra

Every sample point has probability $m_{p} (x) = p^{t (x)} (1 - p)^{n - t (x)} > 0$ , so the common support is the whole finite sample space. Counting measure assigns mass one to each point, so the densities equal these point masses; for $x, y$ , their ratio is $m_{p} (x) / m_{p} (y) = [p / (1 - p)]^{t (x) - t (y)}$ .

2.1A1F1F2step 1.1givenalgebra∎

If $t (x) = t (y)$ , this ratio is one for all $p$ . If $d = t (x) - t (y) \neq 0$ , its values at $p = 1 / 2$ and $p = 1 / 3$ are respectively $1$ and $(1 / 2)^{d}$ , which differ. Thus equivalence classes are exactly the level sets of $T$ . By [F2], $T$ is minimal sufficient.

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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)
Jiahao Tian, Chapter 2: Sufficiency and Exponential Families (standard reference, not scraped)