Testing multiple inequality hypotheses: A smoothed indicator approach

This paper proposes a class of origin-smooth approximators of indicators underlying the sum-of-negative-part statistic for testing multiple inequalities. The need for simulation or bootstrap to obtain test critical values is thereby obviated. A simple procedure is enabled using fixed critical values. The test is shown to have correct asymptotic size in the uniform sense that supremum finite-sample rejection probability over null-restricted data distributions tends asymptotically to nominal significance level. This applies under weak assumptions allowing for estimator covariance singularity. The test is unbiased for a wide class of local alternatives. A new theorem establishes directions in which the test is locally most powerful. The proposed procedure is compared with predominant existing tests in structure, theory and simulation.