A Test in the Presence of Nuisance Parameters

Abstract We are interested in testing Ψ = 0 against an alternative in the presence of some nuisance parameter λ. The usual procedure for such problems is to use a test statistic that is a function of the data only. Let q(λ) denote the p-value at a given value λ. If q(λ) does not depend on λ, then in principle we can apply this procedure. However, a major difficulty that arises in many situations is that q(λ) depends on λ and therefore cannot be used as a p-value. In such cases, the usual approach is to define the p-value as the supremum of q(λ) over the nuisance parameter space. Because this approach ignores sample information about λ, it may be unnecessarily conservative; this is a serious problem in order restricted inference. To overcome this, I propose the following. Obtain, say, a 99% confidence region for λ under the null hypothesis. Now, for a given λ, let T(λ) be a test statistic and r(λ) be the p-value. The test procedure is to reject the null hypothesis if {0.01 + supremum of r(λ) over the 99% c...