A smoothed p-value test when there is a nuisance parameter under the alternative

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We present a new test when there is a nuisance parameter λ under the alternative hypothesis. The test exploits the p-value occupation time [PVOT], the measure of the subset of λ on which a p-value test based on a test statistic Tn(λ) rejects the null hypothesis. Key contributions are: (i) An asymptotic critical value upper bound for our test is the significance level α, making inference easy. (ii) We only require Tn(λ) to have a known or bootstrappable limit distribution, hence we do not require n-Gaussian asymptotics, allowing for weak or non-identification, boundary values, heavy tails, infill asymptotics, and so on. (iii) A test based on the transformed p-value supλ∈Λpn(λ) may be conservative and in some cases have trivial power, while the PVOT naturally controls for this by smoothing over the nuisance parameter space. Finally, (iv) the PVOT uniquely allows for bootstrap inference in the presence of nuisance parameters when some estimated parameters may not be identified.