p-values and measure theory

Self-reassurance that p-value properties don’t depend on regularity assumptions on the test statistic.

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June 7, 2023

Let (Ω,E,Pr) be a probability space, where Ω is the space of random outcomes, E the σ-algebra of measurable events, and P the probability measure.

Given a random variable T:ΩR, define pT:Ω[0,1] as:

pT(ω)=Pr({ωΩ|T(ω)T(ω)})

Theorem. pT is measurable and Pr(pTα)α for all α[0,1]. Equality holds if and only if there exists a sequence {ωn}nN such that pT(ωn)α, and pT(ωn)α as n.

Proof. Let α[0,1], and denote: ET(ω)={ωΩ|T(ω)T(ω)}, so that pT(ω)=Pr(ET(ω)).

Assume first that there exists ωαpT1(α), that is to say Pr(ET(ω))=α. We can show that:

NT(ωα)={ω|pT(ω)α}ET(ωα) is a measurable, zero probability set, which proves the thesis for this particular case. As a matter of fact, for any ωΩ, if pT(ω)α and T(ω)<T(ωα), then we must have:

Pr({ωΩ|T(ωα)>T(ω)T(ω)})=pT(ω)α=0. If t=infpT(ω)αT(ω) and {an}nN is a sequence in Ω such that T(an)t as n, then:

NT(ωα)n{ωΩ|T(ωα)>T(ω)T(an)}, the right hand side being a probability zero set.

If pT1(α) is empty, let α=suppT(ω)αp(ω), and let {bn}nN be a sequence in Ω such that pT(bn)α as n. Then:

{ω|pT(ω)α}={ω|pT(ω)α}=n{ω|pT(ω)pT(bn)}, so that, from the particular case proved earlier, we have:

Pr(pTα)=limnPr(pTpT(bn))limnpT(bn)=αα, as was to be proved.

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BibTeX citation:
@online{gherardi2023,
  author = {Gherardi, Valerio},
  title = {P-Values and Measure Theory},
  date = {2023-06-07},
  url = {https://vgherard.github.io/posts/2023-06-07-p-values-and-measure-theory/},
  langid = {en}
}
For attribution, please cite this work as:
Gherardi, Valerio. 2023. “P-Values and Measure Theory.” June 7, 2023. https://vgherard.github.io/posts/2023-06-07-p-values-and-measure-theory/.