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Under suitable assumptions we prove that there does not exist a perfect exact multiple test procedure that would apply simultaneously to any positive correlation coefficient even with a known distribution of test statistics. This nonexistence theorem holds for all simple tests under normal distribution, and holds for all tests under Ferguson’s distribution. Given the nonexistence of a perfect exact test, we provide least conservative tests using three parametric models. The average conservativeness of these tests can be reduced to as low as 1/8 of that of the widely used Simes test assuming normal distribution. Power analysis indicates that the newly proposed tests are useful in practice.

Least Conservative Critical Boundaries of Multiple Hypothesis Testing in a Range of Correlation Values