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% confidence region for λ is less than the nominal level such as 0. 05. In contrast to the usual procedure, an attractive feature of this procedure is that it allows us to choose a test statistic as a function of λ. A data example is used to illustrate the procedure in a simulation study I observed that this test performed better than the traditional conservative procedure. Although this approach was originally developed for order restricted inference problems, the main results have wide applicability.