Why is the duality useful in practice?

It lets you build a confidence interval whenever you can test hypotheses, even with no pivot or closed form, by collecting all parameter values the test does not reject; profile-likelihood intervals are a common example.

Does the duality mean tests and intervals always agree?

Yes, by construction: a value lies outside the confidence interval exactly when the corresponding null hypothesis is rejected at the matching level, so the two reach the same conclusion.

Duality of Tests and Confidence Sets

Every confidence set corresponds to a family of hypothesis tests and vice versa: the parameter values a test does not reject form a confidence set at the complementary level.

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Definition

The duality of tests and confidence sets is the equivalence by which the set of parameter values not rejected by a family of level-alpha tests is a confidence set of coverage one minus alpha, and any confidence set defines such a family of tests.

Scope

This topic covers the formal correspondence between acceptance regions of level-alpha tests and confidence sets of level one minus alpha, the construction of confidence sets by test inversion, the transfer of optimality so that uniformly most powerful unbiased tests yield uniformly most accurate unbiased confidence sets, the resulting one-sided and two-sided intervals, and the use of inversion when no convenient pivot exists.

Core questions

How does the acceptance region of a test, read as a function of the parameter, define a confidence set?
Why does the coverage of the inverted set equal one minus the size of the tests?
How does the optimality of a test transfer to accuracy of the corresponding confidence set?
When is test inversion preferable to the pivotal method?

Key theories

Test inversion: Fixing the data and collecting all parameter values whose test accepts the data produces a confidence set whose coverage is one minus the common size of the tests.
Uniformly most accurate confidence sets: Inverting a uniformly most powerful unbiased test yields a confidence set that minimizes the probability of covering false parameter values, the confidence analogue of optimal power.

Clinical relevance

Test inversion is the practical route to confidence intervals when no closed-form pivot exists, for example profile-likelihood intervals for odds ratios and hazard ratios, which are obtained by collecting the parameter values a likelihood-ratio test would not reject.

History

Neyman's 1937 confidence theory already exhibited the link between intervals and tests, and Lehmann's optimality theory of tests, later revised with Romano, made the transfer of optimality to confidence sets explicit and systematic.

Key figures

Jerzy Neyman
Erich L. Lehmann
Joseph P. Romano
George Casella

Seminal works

lehmannRomano2005

Frequently asked questions

Why is the duality useful in practice?: It lets you build a confidence interval whenever you can test hypotheses, even with no pivot or closed form, by collecting all parameter values the test does not reject; profile-likelihood intervals are a common example.
Does the duality mean tests and intervals always agree?: Yes, by construction: a value lies outside the confidence interval exactly when the corresponding null hypothesis is rejected at the matching level, so the two reach the same conclusion.