Combining asymptotically normal tests: case studies in comparison of two groups

When several candidate tests are available for a given testing problem, and each has nice properties with respect to different criteria such as efficiency and robustness, it is desirable to combine them. We discuss various combined tests based on asymptotically normal tests. When the means of two standardized tests under contiguous alternatives are close, we show that the maximum of the two tests appears to have an overall best performance compared with other forms of combined tests considered, and that it retains most power compared with the better one of the two tests combined. As an application, for testing zero location shift between two groups, we studied the normal, Wilcoxon, median tests and their combined tests. Because of their structural differences, the joint convergence and the asymptotic correlation of the tests are not easily derived from the usual asymptotic arguments of the tests. We developed a novel application of martingale theory to obtain the asymptotic correlations and their estimators. Simulation studies were also performed to examine the small sample properties of these combined tests. Finally we illustrate the methods by a real data example.