The Annals of Statistics

Dependency and false discovery rate: Asymptotics

Helmut Finner, Thorsten Dickhaus, and Markus Roters

Full-text: Open access


Some effort has been undertaken over the last decade to provide conditions for the control of the false discovery rate by the linear step-up procedure (LSU) for testing $n$ hypotheses when test statistics are dependent. In this paper we investigate the expected error rate (EER) and the false discovery rate (FDR) in some extreme parameter configurations when $n$ tends to infinity for test statistics being exchangeable under null hypotheses. All results are derived in terms of $p$-values. In a general setup we present a series of results concerning the interrelation of Simes’ rejection curve and the (limiting) empirical distribution function of the $p$-values. Main objects under investigation are largest (limiting) crossing points between these functions, which play a key role in deriving explicit formulas for EER and FDR. As specific examples we investigate equi-correlated normal and $t$-variables in more detail and compute the limiting EER and FDR theoretically and numerically. A surprising limit behavior occurs if these models tend to independence.

Article information

Ann. Statist., Volume 35, Number 4 (2007), 1432-1455.

First available in Project Euclid: 29 August 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J15: Paired and multiple comparisons 62F05: Asymptotic properties of tests
Secondary: 62F03: Hypothesis testing 60F99: None of the above, but in this section

exchangeable test statistics expected error rate false discovery rate Glivenko–Cantelli theorem largest crossing point least favorable configurations multiple comparisons multiple test procedure multivariate total positivity of order 2 positive regression dependency Simes’ test


Finner, Helmut; Dickhaus, Thorsten; Roters, Markus. Dependency and false discovery rate: Asymptotics. Ann. Statist. 35 (2007), no. 4, 1432--1455. doi:10.1214/009053607000000046.

Export citation


  • Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289–300.
  • Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Ann. Statist. 29 1165–1188.
  • Eklund, G. and Seeger, P. (1965). Massignifikansanalys. Statistisk Tidskrift Stockholm 3 355–365.
  • Finner, H., Dickhaus, T. and Roters, M. (2008). Asymptotic tail properties of Student's $t$-distribution. Comm. Statist. Theory Methods 37. To appear.
  • Finner, H. and Roters, M. (1998). Asymptotic comparison of step-down and step-up multiple test procedures based on exchangeable test statistics. Ann. Statist. 26 505–524.
  • Finner, H. and Roters, M. (2001). Asymptotic sharpness of product-type inequalities for maxima of random variables with applications in multiple comparisons. J. Statist. Plann. Inference 98 39–56.
  • Finner, H. and Roters, M. (2001). On the false discovery rate and expected number of type I errors. Biom. J. 43 985–1005.
  • Finner, H. and Roters, M. (2002). Multiple hypotheses testing and expected number of type I errors. Ann. Statist. 30 220–238.
  • Genovese, C. R. and Wasserman, L. (2002). Operating characteristics and extensions of the false discovery rate procedure. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 499–517.
  • Genovese, C. R. and Wasserman, L. (2004). A stochastic process approach to false discovery control. Ann. Statist. 32 1035–1061.
  • Karlin, S. (1968). Total Positivity 1. Stanford Univ. Press.
  • Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10 467–498.
  • Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. Ann. Statist. 30 239–257.
  • Seeger, P. (1966). Variance Analysis of Complete Designs. Some Practical Aspects. Almqvist and Wiksell, Uppsala.
  • Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. Biometrika 73 751–754.
  • Storey, J. D., Taylor, J. E. and Siegmund, D. (2004). Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: A unified approach. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 187–205.