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Conditional level implies unconditional level

Statement

Let $φ$ be a test for nonempty disjoint null and alternative parameter sets $Θ_{0}, Θ_{1}$ as in Test size and power, let $S$ be a statistic, and let $α \in [0, 1]$ . If for every $θ \in Θ_{0}$ there is a conditional-expectation version $q_{θ} = E_{θ} [φ (X) ∣ σ (S)]$ with $q_{θ} \leq α$ $P_{θ}$ -almost surely, then $E_{θ} [φ (X)] \leq α$ for every $θ \in Θ_{0}$ . Consequently, the test has size at most $α$ , where size is the supremum of its rejection expectation over the null family as in Test size and power.

Facts & Assumptions

Given: A test $φ$ , null family $Θ_{0}$ , statistic $S$ , and conditional level bound $α$ .

[F1]

Conditional level at most $α$ means that for each null parameter a supplied conditional-expectation version $q_{θ}$ satisfies $q_{θ} \leq α$ almost surely. Conditional tests given a statistic

[F2]

A conditional-expectation version satisfies $\int_{A} q_{θ} d P_{θ} = \int_{A} φ d P_{θ}$ for every $A \in σ (S)$ ; in particular $Ω \in σ (S)$ . Conditional expectation given a sigma algebra

[F3]

For integrable real random variables, almost-sure pointwise order implies the same order of expectations. Linearity, monotonicity, and the modulus bound for expectation

[A1]

The test size convention is $\sup_{θ \in Θ_{0}} E_{θ} φ (X)$ as in Test size and power.

Proof

1.1F1F2givenchoose

Fix $θ \in Θ_{0}$ and choose the conditional version $q_{θ}$ supplied by [F1]. Taking $A = Ω$ in [F2] gives $E_{θ} q_{θ} = E_{θ} φ (X)$ .

2.1F3step 1.1algebra

Since $q_{θ} \leq α$ almost surely and both variables are integrable, [F3] gives $E_{θ} q_{θ} \leq E_{θ} α = α$ . Combining with step 1.1 yields $E_{θ} φ (X) \leq α$ .

3.1A1step 2.1∎

The argument holds for every $θ \in Θ_{0}$ , so taking the supremum of these rejection expectations gives size at most $α$ under the convention [A1]. No common exceptional null set across parameter values is needed.

Depends on

Used by

Fisher's exact test from two Bernoulli samples Example

Dependency tree · two levels

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Sources

George Deligiannidis, Foundations of Statistical Inference, Chapter 13, PDF pp. 77–78 (standard reference, not scraped)
Berkeley Stat 210A Course Reader, Testing with Nuisance Parameters, Conditional testing section (standard reference, not scraped)