A smoothed \(p\)-value test when there is a nuisance parameter under the alternative. (English) Zbl 1522.62023
Summary: We present a new test when there is a nuisance parameter \(\lambda\) under the alternative hypothesis. The test exploits the \(p\)-value occupation time [PVOT], the measure of the subset of \(\lambda\) on which a \(p\)-value test based on a test statistic \(\mathcal{I}_n(\lambda)\) rejects the null hypothesis. Key contributions are:
- (i)
- An asymptotic critical value upper bound for our test is the significance level \(\alpha\), making inference easy.
- (ii)
- We only require \(\mathcal{I}_n(\lambda)\) to have a known or bootstrappable limit distribution, hence we do not require \(\sqrt{n}\)-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 \(\sup_{\lambda\in\Lambda}p_n(\lambda)\) 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.
MSC:
| 62G10 | Nonparametric hypothesis testing |
| 62M99 | Inference from stochastic processes |
| 62F35 | Robustness and adaptive procedures (parametric inference) |
Keywords:
\(p\)-value test; empirical process test; nuisance parameter; weak identification; GARCH test; omitted nonlinearity testReferences:
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