Hypothesis testing with a restricted parameter space

This paper considers hypothesis tests when the parameter space is restricted under the alternative hypothesis. Multivariate one-sided tests are a leading example. Optimal tests, called directed tests, are derived using a weighted average power criterion. Exact results are established first for Gaussian linear regression models with known variance. Asymptotic analogues are then established for dynamic nonlinear models. Simulation is used to compare the tests discussed in the paper. The D-W∞ directed test is found to perform best in an overall sense for multivariate one-sided alternatives with the likelihood ratio test being a close second. The D-W∞ and likelihood ratio tests are found to perform best for mixed one- and two-sided alternatives.