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Countable factorization criterion for sufficiency

Statement

Let $X$ be a nonempty countable set with its full $σ$ -field, let $Θ \neq \emptyset$ , let $T : X \to Y$ be a statistic into a standard Borel space, and write $p_{θ} (x) = P_{θ} ({x})$ . Then $T$ is sufficient if and only if there are nonnegative functions $h : X \to [0, \infty)$ and Borel $g_{θ} : Y \to [0, \infty)$ such that

$p_{θ} (x) = g_{θ} (T x) h (x) (θ \in Θ, x \in X) .$

The pointwise convention is $g_{θ} (t) = 0$ whenever $M (t) = 0$ , where $M (t) = \sup_{θ} P_{θ} (T = t)$ on the countable image $T (X)$ ; outside that image set $g_{θ} = 0$ .

Facts & Assumptions

Given: the countable experiment, the statistic $T$ , and the displayed factorization or a common kernel, according to the implication being proved.

[F1]

Sufficient statistics: sufficiency means one probability kernel $K$ satisfying the conditional-law identity for every model law.

[F2]

Measure kernel and probability kernel: for each target event, a probability kernel is measurable in its conditioning argument and is a probability measure in its event argument.

[F3]

Standard Borel spaces: singletons in a standard Borel space are Borel, and countable subsets are Borel.

Proof

1.1F1F2F3givenconstruct

Suppose $T$ is sufficient and fix a common kernel $K$ from [F1]. For $t \in T (X)$ set $m_{θ} (t) = P_{θ} (T = t)$ and $M (t) = \sup_{θ} m_{θ} (t)$ . The image is countable by countability of $X$ , hence Borel by [F3]; every function on this image extended by zero is Borel. Define $h (x) = M (T x) K (T x, {x})$ and define $g_{θ} (t) = m_{θ} (t) / M (t)$ when $M (t) > 0$ , and zero when $M (t) = 0$ . Then $g_{θ}$ is Borel and nonnegative.

2.1F1step 1.1givenalgebra

If $M (T x) > 0$ , choose $θ_{0}$ with $m_{θ_{0}} (T x) > 0$ . The conditional identity in [F1] gives $K (T x, T^{- 1} {T x}) = 1$ and $p_{θ} (x) = m_{θ} (T x) K (T x, {x})$ for every $θ$ ; if $m_{θ} (T x) = 0$ , both sides vanish by the same identity. Thus $p_{θ} (x) = g_{θ} (T x) h (x)$ . If $M (T x) = 0$ , every law gives that fiber probability zero, so $p_{θ} (x) = h (x) = g_{θ} (T x) = 0$ . This proves the forward implication pointwise.

2.2F2F3givenchooseconstructstep 1.1

Conversely suppose the factorization holds. For each $t \in T (X)$ let $F_{t} = T^{- 1} {t}$ and $H (t) = \sum_{x \in F_{t}} h (x)$ . If some $m_{θ} (t) > 0$ , then $g_{θ} (t) > 0$ and $0 < H (t) = m_{θ} (t) / g_{θ} (t) < \infty$ , since the sum of the probabilities on the fiber is at most one. In that case put $K (t, {x}) = h (x) / H (t)$ on $F_{t}$ and zero off $F_{t}$ . If all $m_{θ} (t) = 0$ , choose the first point of $F_{t}$ in a fixed enumeration and let $K (t, \cdot)$ be its point mass. For $t \notin T (X)$ choose one fixed $x_{0} \in X$ and set $K (t, \cdot) = δ_{x_{0}}$ . The countable image and enumeration make every coordinate $t \mapsto K (t, {x})$ Borel. For each $A \subseteq X$ , $K (t, A) = \sum_{x \in A} K (t, {x})$ is a countable sum of Borel functions, so it is Borel; the point-mass assignments are countably additive probability measures.

3.1F1step 2.2givenalgebra∎

On an active fiber, for every $A \subseteq X$ , $P_{θ} (A \cap F_{t}) = g_{θ} (t) \sum_{x \in A \cap F_{t}} h (x) = m_{θ} (t) K (t, A)$ . On an inactive fiber both sides are zero. Summing over the countably many $t \in C \cap T (X)$ gives the conditional-law identity for every Borel $C \subseteq Y$ . Thus [F1] holds and $T$ is 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)
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)