Some results on the control of the false discovery rate under dependence

Controlling the false discovery rate (FDR) is a powerful approach to multiple testing, with procedures developed with applications in many areas. Dependence among the test statistics is a common problem, and many attempts have been made to extend the procedures. In this paper, we show that a certain degree of dependence is allowed among the test statistics, when the number of tests is large, with no need for any correction. We then suggest a way to conservatively estimate the proportion of false nulls, both under dependence and independence, and discuss the advantages of using such estimators when controlling the FDR.