## The Annals of Statistics

### Dependency and false discovery rate: Asymptotics

#### Abstract

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

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

Dates
First available in Project Euclid: 29 August 2007

https://projecteuclid.org/euclid.aos/1188405617

Digital Object Identifier
doi:10.1214/009053607000000046

Mathematical Reviews number (MathSciNet)
MR2351092

Zentralblatt MATH identifier
1125.62076

#### Citation

Finner, Helmut; Dickhaus, Thorsten; Roters, Markus. Dependency and false discovery rate: Asymptotics. Ann. Statist. 35 (2007), no. 4, 1432--1455. doi:10.1214/009053607000000046. https://projecteuclid.org/euclid.aos/1188405617

#### References

• 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.