Hypothesis testing when a nuisance parameter is present only under the alternative

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Abstract
Suppose that the distribution of a random variable representing the outcome of an experiment depends on two parameters ξ and θ and that we wish to test the hypothesis ξ = 0 against the alternative ξ > 0. If the distribution does not depend on θ when ξ = 0, standard asymptotic methods such as likelihood ratio testing or C(α) testing are not directly applicable. However, these methods may, under appropriate conditions, be used to reduce the problem to one involving inference from a Gaussian process. This simplified problem is examined and a test which may be derived as a likelihood ratio test or from the union-intersection principle is introduced. Approximate expressions for the significance level and power are obtained.

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