Source: Electron. J. Statist.
We investigate the performance of a family of multiple comparison procedures for strong control of the False Discovery Rate (). The is the expected False Discovery Proportion (), that is, the expected fraction of false rejections among all rejected hypotheses. A number of refinements to the original Benjamini-Hochberg procedure  have been proposed, to increase power by estimating the proportion of true null hypotheses, either implicitly, leading to one-stage adaptive procedures [4, 7] or explicitly, leading to two-stage adaptive (or plug-in) procedures [2, 21].
We use a variant of the stochastic process approach proposed by Genovese and Wasserman  to study the fluctuations of the achieved with each of these procedures around its expectation, for independent tested hypotheses.
We introduce a framework for the derivation of generic Central Limit Theorems for the of these procedures, characterizing the associated regularity conditions, and comparing the asymptotic power of the various procedures. We interpret recently proposed one-stage adaptive procedures [4, 7] as fixed points in the iteration of well known two-stage adaptive procedures [2, 21].
 Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing., J. R. Stat. Soc. Ser. B Stat. Methodol. 57, 1, 289–300.
 Benjamini, Y., Krieger, A. M., and Yekutieli, D. (2006). Adaptive linear step-up procedures that control the false discovery rate., Biometrika 93, 3, 491–507.
 Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency., Ann. Statist. 29, 4, 1165–1188.
 Blanchard, G. and Roquain, E. (2008). Adaptive FDR control under independence and dependence. arXiv preprint, math.ST/0707.0536v2.
 Chi, Z. (2007). On the performance of FDR control: constraints and a partial solution., Ann. Statist. 35, 4, 1409–1431.
 Donsker, M. D. (1951). An invariance principle for certain probability limit theorems., Mem. Amer. Math. Soc. 6, 12.
Mathematical Reviews (MathSciNet): MR40613
 Finner, H., Dickhaus, T., and Roters, M. (2008). On the false discovery rate and an asymptotically optimal rejection curve., Ann. Statist. (to appear).
 Finner, H. and Roters, M. (2001). On the false discovery rate and expected type I errors., Biom. J. 43, 8, 985–1005.
 Finner, H. and Roters, M. (2002). Multiple hypotheses testing and expected number of type I errors., Ann. Statist. 30, 1, 220–238.
 Genovese, C. and Wasserman, L. (2002). Operating characteristics and extensions of the false discovery rate., J. R. Stat. Soc. Ser. B Stat. Methodol. 64, 3, 499–517.
 Genovese, C. and Wasserman, L. (2004). A stochastic process approach to false discovery control., Ann. Statist. 32, 3, 1035–1061.
 Genovese, C. and Wasserman, L. (2006). Exceedance control of the false discovery proportion., J. Amer. Statist. Assoc. 101, 476, 1408–1417.
 Lehmann, E. L. and Romano, J. P. (2005). Generalizations of the familywise error rate., Ann. Statist 33, 3, 1138–1154.
 Neuvial, P. (2008). Intrinsic bounds and false discovery rate control in multiple testing problems. In revision for, J. Mach. Learn. Res.
 Perone Pacifico, M., Genovese, C., Verdinelli, I., and Wasserman, L. (2004). False discovery control for random fields., J. Amer. Statist. Assoc. 99, 468, 1002–1014.
 Romano, J. P. and Shaikh, A. M. (2006a). On stepdown control of the false discovery proportion. In, Optimality: The Second Erich L. Lehmann Symposium, J. Rojo, Ed. Vol. 33. Institute of Mathematical Statistics, Beachwood, Ohio, USA.
 Romano, J. P. and Shaikh, A. M. (2006b). Step-up procedures for control of generalizations of the family-wise error rate., Ann. Statist. 34, 4, 1850–1873.
 Romano, J. P. and Wolf, M. (2007). Control of generalized error rates in multiple testing., Ann. Statist. 35, 4, 1378–1408.
 Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures., Ann. Statist. 30, 1, 239–257.
 Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance., Biometrika 73, 3, 751-754.
Mathematical Reviews (MathSciNet): MR897872
 Storey, J. D. (2002). A direct approach to false discovery rates., J. R. Stat. Soc. Ser. B Stat. Methodol. 64, 3, 479–498.
 Storey, J. D. (2003). The positive false discovery rate: a bayesian interpretation and the q-value., Ann. Statist. 31, 6, 2013–2035.
 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, 1, 187–205.
 Storey, J. D. and Tibshirani, R. (2003). SAM thresholding and false discovery rates for detecting differential gene expression in DNA microarrays. In, The Analysis of Gene Expression Data: Methods and Software. Springer, New York, 272–290.
 Sun, W. and Cai, T. T. (2007). Oracle and adaptive compound decision rules for false discovery rate control., J. Amer. Statist. Assoc. 102, 479, 901–912.
 van der Laan, M. J., Dudoit, S., and Pollard, K. S. (2004). Augmentation procedures for control of the generalized family-wise error rate and tail probabilities for the proportion of false positives., Stat. Appl. Genet. Mol. Biol. 3, Art. 15, 27 pp.
 van der Vaart, A. W. (1998)., Asymptotic Statistics. Cambridge University Press.
 Wu, W. B. (2008). On false discovery control under dependence., Ann. Statist. 36, 1, 364–380.