Asymptotic tests of composite hypotheses

Test statistics that are suitable for testing composite hypotheses are typically non-pivotal, and conservative bounds are commonly used to test composite hypotheses. In this paper, we propose a testing procedure for composite hypotheses that incorporates additional sample information. This avoids, as n→∞, the use of conservative bounds and leads to tests with better power than standard tests. The testing procedure satisfies a novel similarity condition that is relevant for asymptotic tests of composite hypotheses, and we show that this is a necessary condition for a test to be unbiased. The procedure is particularly useful for simultaneous testing of multiple inequalities, in particular when the number of inequalities is large. This is the situation for the multiple comparisons of forecasting models, and we show that the new testing procedure dominates the ‘reality check’ of White (2000) and avoids certain pitfalls.