How Much Can Gaussian Dependence Inflate the Benjamini-Hochberg Procedure's FDR?

2026/07/16 by Lihua Lei
#math.ST #stat.TH

paper · pdf

Abstract

We study the worst-case false discovery rate (FDR) of the Benjamini-Hochberg procedure for both one- and two-sided Gaussian tests when the correlation matrix is otherwise unrestricted. In each setting we construct a q-indexed family of finite Gaussian models whose FDR divided by q diverges as q\downarrow0, disproving any universal multiplicative FDR bound. For two-sided tests, the supremum over the number of hypotheses, mean vector, and correlation matrix is at least an explicit ℓ₌(q)>q satisfying ℓ₌(q)=(q√(log(1/q)))/(2√π)+c_ℓ q+o(q), c_ℓ=0.6492828…. For the one-sided hypotheses H_i:θ_i≤0, a sign-reversed one-common-factor construction gives the stronger explicit lower bound ℓ_≤(q)>q, with ℓ_≤(q)=(q√(log(1/q)))/(√π) +\frac q2+o(q). Finally, we prove an O\q√(log(1/q))\ upper bound for the two-sided one-common-factor class and the matching upper bound q√(log(1/q))/√π+O(q) for the one-sided one-common-factor class.

Related

• On the false discovery proportion convergence under Gaussian equi-correlation
• Dependence-Aware False Discovery Rate Control in Two-Sided Gaussian Mean Testing
• Observation on F.W.E.R. and F.D.R. for correlated normal
• Finite sample bounds for expected number of false rejections under martingale dependence with applications to FDR
• The False Discovery Rate (FDR) of Multiple Tests in a Class Room Lecture