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

Abstract

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.

Keywords:

• Admissible curve
• Dependence
• Multiple test procedure
• Relative conservativeness
• Simes test

Acknowledgments

We thank the associate editor and two referees for their comments which helped to improve the article.