TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Two-stage strategy-specific effect estimators are design-unbiased

Statement

Under the two-stage design of Two-stage randomized design and strategy-specific effects, for either strategy $s \in {ϕ, ψ}$ , treatment value $z \in {0, 1}$ , and group $i$ , $E [{\hat{\overset{ˉ}{Y}}}_{i} (z; s) ∣ S_{i} = s] = {\overset{ˉ}{Y}}_{i} (z; s), E [{\hat{\overset{ˉ}{Y}}}_{i} (s) ∣ S_{i} = s] = {\overset{ˉ}{Y}}_{i} (s) .$ Consequently, $E [\hat{\overset{ˉ}{Y}} (z; s)] = \overset{ˉ}{Y} (z; s), E [\hat{\overset{ˉ}{Y}} (s)] = \overset{ˉ}{Y} (s) .$ By taking the displayed contrasts, the design estimators of $D E (ψ)$ , $I E (ϕ, ψ)$ , $T E (ϕ, ψ)$ , and $O E (ϕ, ψ)$ are unbiased. The proof does not require stratified interference or additive effects.

Facts & Assumptions

Given: The finite two-stage assignment law, fixed potential outcomes under partial interference, and the group and population targets and estimators in Two-stage randomized design and strategy-specific effects.

[F1]

Two-stage randomized design and strategy-specific effects: exactly $C_{s}$ groups are assigned to each strategy $s$ , and within group $i$ under $s$ exactly $K_{i}^{s}$ of its $n_{i}$ units are chosen uniformly for treatment; the estimators divide by the fixed treatment-arm sizes and fixed group counts.

[F2]

Finite simple-random-sampling moment identities: under uniform fixed-size sampling from $n$ units, each unit's inclusion probability is $m / n$ , and the sample mean of fixed values has expectation equal to their finite-population mean.

[F3]

Finite probability spaces, outcome weights, events, and event probabilities and Expectation of a real random variable on a finite probability space: all assignment laws here are finite, and their expectations are finite weighted sums.

[F4]

Conditional expectation given a sigma algebra: a conditional-expectation version is measurable with respect to the conditioning sigma-algebra and has the same integrals on its events.

[F5]

Statistics and estimators: a statistic with the target space of an estimand is an estimator of that estimand.

[F6]

Estimands on a statistical model: an estimand is a specified function from the model's parameter set to its target-value set.

[F7]

Laws of finite sums and finite products: finite sums are additive and scaling distributes over finite sums.

[F8]

Conditional expectation on positive-probability events in finite spaces: positive-probability event conditioning is the finite conditional weighted sum, and averaging conditional means over a finite partition recovers unconditional expectation.

[F9]

Expectation is linear for every finite family of random variables, without any independence hypothesis: expectation distributes over finite real linear combinations on a finite probability space.

Proof

1.1F1F3F4F7F8F9given

Fix $i, s, z$ . The number $m_{i z}^{s}$ of units with $Z_{i j} = z$ is fixed and positive under [F1]. Each of the $n_{i}$ units has conditional probability $m_{i z}^{s} / n_{i}$ of receiving $z$ . Expanding the positive-probability event conditional mean by [F8] and distributing the finite sum over units by [F9] gives $E [{\hat{\overset{ˉ}{Y}}}_{i} (z; s) ∣ S_{i} = s] = \frac{1}{m_{i z}^{s}} \sum_{j = 1}^{n_{i}} P (Z_{i j} = z ∣ S_{i} = s) E [Y_{i j}^{o b s} ∣ S_{i} = s, Z_{i j} = z] = \frac{1}{n_{i}} \sum_{j = 1}^{n_{i}} E_{i, s} [Y_{i j} (Z_{i}) ∣ Z_{i j} = z] = {\overset{ˉ}{Y}}_{i} (z; s) .$ All conditioning events have positive probability by the bounds on $K_{i}^{s}$ .

1.2F1F3F7F9given

By [F3], expectation linearity [F9] and the definition of ${\overset{ˉ}{Y}}_{i} (s)$ give $E [{\hat{\overset{ˉ}{Y}}}_{i} (s) ∣ S_{i} = s] = \frac{1}{n_{i}} \sum_{j = 1}^{n_{i}} E_{i, s} [Y_{i j} (Z_{i})] = {\overset{ˉ}{Y}}_{i} (s) .$ This uses no restriction on how one unit's outcome depends on other assignments within its group.

2.1step 1.1step 1.2F1F3F7F8F9algebra

Condition on the complete first-stage vector $S$ . The denominators $C_{s}$ are fixed, so expectation linearity [F9] and steps 1.1 and 1.2 yield $E [\hat{\overset{ˉ}{Y}} (z; s) ∣ S] = \frac{1}{C_{s}} \sum_{i : S_{i} = s} {\overset{ˉ}{Y}}_{i} (z; s), E [\hat{\overset{ˉ}{Y}} (s) ∣ S] = \frac{1}{C_{s}} \sum_{i : S_{i} = s} {\overset{ˉ}{Y}}_{i} (s) .$

3.1step 2.1F2F3F7F8F9

The set of groups assigned strategy $s$ is a uniform $C_{s}$ -element subset of the $N$ fixed groups. By [F2], each group has inclusion probability $C_{s} / N$ , so the expectation of either average on the right of step 2.1 is the corresponding target population average. The finite-partition identity [F8] and expectation linearity [F9] prove the two population-mean identities.

4.1step 3.1F3F5F6F9algebra∎

Each effect estimator is the same finite linear contrast of estimated means as its target is of the corresponding target means. Distributing expectation over each contrast by [F9] and applying step 3.1 proves unbiasedness of all four effects. By [F5] and [F6], these statistic-valued contrasts are estimators of the displayed estimands.

Remarks

The assignment space and every sum in this proof are finite. The argument is choice-free and uses no conditional-expectation existence theorem on general probability spaces.

Depends on

Used by

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Michael G. Hudgens and M. Elizabeth Halloran, Toward Causal Inference With Interference, Theorems 1–3 and Appendices A.1–A.3 (standard reference, not scraped)