We consider hypothesis testing problems in which a nuisance parameter is present only under the alternative hypothesis. Standard asymptotic tests, such as likelihood ratio, Lagrange multiplier and Wald tests, are difficult to apply because of problems incurred in obtaining their asymptotic distributions. To overcome this difficulty, we reparameterize the testing problem to one for which an exact small sample test can be constructed using existing hypothesis testing procedures. The reparameterization technique is applied to two examples from the econometrics literature, and an empirical power comparison shows that our test has better power properties than tests previously proposed in the literature. Further, p-values for our test can be computed in 0(n) operations so the test can be implemented efficiently.