{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T21:56:10Z","timestamp":1791582970160,"version":"4.3.4"},"reference-count":36,"publisher":"Institute of Mathematical Statistics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2011,2,1]]},"abstract":"<jats:p>In a context of multiple hypothesis testing, we provide several new exact calculations related to the false discovery proportion (FDP) of step-up and step-down procedures. For step-up procedures, we show that the number of erroneous rejections conditionally on the rejection number is simply a binomial variable, which leads to explicit computations of the c.d.f., the sth moment and the mean of the FDP, the latter corresponding to the false discovery rate (FDR). For step-down procedures, we derive what is to our knowledge the first explicit formula for the FDR valid for any alternative c.d.f. of the p-values. We also derive explicit computations of the power for both step-up and step-down procedures. These formulas are \u201cexplicit\u201d in the sense that they only involve the parameters of the model and the c.d.f. of the order statistics of i.i.d. uniform variables. The p-values are assumed either independent or coming from an equicorrelated multivariate normal model and an additional mixture model for the true\/false hypotheses is used. Our approach is then used to investigate new results which are of interest in their own right, related to least\/most favorable configurations for the FDR and the variance of the FDP.<\/jats:p>","DOI":"10.1214\/10-aos847","type":"journal-article","created":{"date-parts":[[2011,2,15]],"date-time":"2011-02-15T15:56:19Z","timestamp":1297785379000},"page":"584-612","source":"Crossref","is-referenced-by-count":30,"title":["Exact calculations for false discovery proportion with application to least favorable configurations"],"prefix":"10.1214","volume":"39","author":[{"given":"Etienne","family":"Roquain","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fanny","family":"Villers","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"[1] Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. <i>J. Roy. Statist. Soc. Ser. B<\/i> <b>57<\/b> 289\u2013300.","DOI":"10.1111\/j.2517-6161.1995.tb02031.x"},{"key":"3","doi-asserted-by":"publisher","unstructured":"[3] Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. <i>Ann. Statist.<\/i> <b>29<\/b> 1165\u20131188.","DOI":"10.1214\/aos\/1013699998"},{"key":"2","doi-asserted-by":"publisher","unstructured":"[2] Benjamini, Y., Krieger, A. M. and Yekutieli, D. (2006). Adaptive linear step-up procedures that control the false discovery rate. <i>Biometrika<\/i> <b>93<\/b> 491\u2013507.","DOI":"10.1093\/biomet\/93.3.491"},{"key":"10","doi-asserted-by":"publisher","unstructured":"[10] Efron, B. (2008). Microarrays, empirical Bayes and the two-groups model. <i>Statist. Sci.<\/i> <b>23<\/b> 1\u201322.","DOI":"10.1214\/07-STS236"},{"key":"12","doi-asserted-by":"publisher","unstructured":"[12] Efron, B., Tibshirani, R., Storey, J. D. and Tusher, V. (2001). Empirical Bayes analysis of a microarray experiment. <i>J. Amer. Statist. Assoc.<\/i> <b>96<\/b> 1151\u20131160.","DOI":"10.1198\/016214501753382129"},{"key":"4","doi-asserted-by":"publisher","unstructured":"[4] Blanchard, G. and Roquain, E. (2008). Two simple sufficient conditions for FDR control. <i>Electron. J. Stat.<\/i> <b>2<\/b> 963\u2013992.","DOI":"10.1214\/08-EJS180"},{"key":"5","unstructured":"[5] Blanchard, G. and Roquain, E. (2009). Adaptive false discovery rate control under independence and dependence. <i>J. Mach. Learn. Res.<\/i> <b>10<\/b> 2837\u20132871."},{"key":"6","doi-asserted-by":"publisher","unstructured":"[6] Chi, Z. (2007). On the performance of FDR control: Constraints and a partial solution. <i>Ann. Statist.<\/i> <b>35<\/b> 1409\u20131431.","DOI":"10.1214\/009053607000000037"},{"key":"7","unstructured":"[7] Chi, Z. and Tan, Z. (2008). Positive false discovery proportions: Intrinsic bounds and adaptive control. <i>Statist. Sinica<\/i> <b>18<\/b> 837\u2013860."},{"key":"9","doi-asserted-by":"publisher","unstructured":"[9] Donoho, D. and Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures. <i>Ann. Statist.<\/i> <b>32<\/b> 962\u2013994.","DOI":"10.1214\/009053604000000265"},{"key":"11","doi-asserted-by":"publisher","unstructured":"[11] Efron, B. (2010). Correlated <i>z<\/i>-values and the accuracy of large-scale statistical estimates. <i>J. Amer. Statist. Assoc.<\/i> <b>105<\/b> 1042\u20131055.","DOI":"10.1198\/jasa.2010.tm09129"},{"key":"13","doi-asserted-by":"publisher","unstructured":"[13] Ferreira, J. A. and Zwinderman, A. H. (2006). On the Benjamini\u2013Hochberg method. <i>Ann. Statist.<\/i> <b>34<\/b> 1827\u20131849.","DOI":"10.1214\/009053606000000425"},{"key":"14","doi-asserted-by":"publisher","unstructured":"[14] Finner, H., Dickhaus, R. and Roters, M. (2009). On the false discovery rate and an asymptotically optimal rejection curve. <i>Ann. Statist.<\/i> <b>37<\/b> 596\u2013618.","DOI":"10.1214\/07-AOS569"},{"key":"15","doi-asserted-by":"publisher","unstructured":"[15] Finner, H., Dickhaus, T. and Roters, M. (2007). Dependency and false discovery rate: Asymptotics. <i>Ann. Statist.<\/i> <b>35<\/b> 1432\u20131455.","DOI":"10.1214\/009053607000000046"},{"key":"16","doi-asserted-by":"publisher","unstructured":"[16] Finner, H. and Roters, M. (2002). Multiple hypotheses testing and expected number of type I errors. <i>Ann. Statist.<\/i> <b>30<\/b> 220\u2013238.","DOI":"10.1214\/aos\/1015362191"},{"key":"17","doi-asserted-by":"publisher","unstructured":"[17] Gavrilov, Y., Benjamini, Y. and Sarkar, S. K. (2009). An adaptive step-down procedure with proven FDR control under independence. <i>Ann. Statist.<\/i> <b>37<\/b> 619\u2013629.","DOI":"10.1214\/07-AOS586"},{"key":"18","doi-asserted-by":"publisher","unstructured":"[18] Genovese, C. and Wasserman, L. (2002). Operating characteristics and extensions of the false discovery rate procedure. <i>J. R. Stat. Soc. Ser. B Stat. Methodol.<\/i> <b>64<\/b> 499\u2013517.","DOI":"10.1111\/1467-9868.00347"},{"key":"19","doi-asserted-by":"publisher","unstructured":"[19] Genovese, C. and Wasserman, L. (2004). A stochastic process approach to false discovery control. <i>Ann. Statist.<\/i> <b>32<\/b> 1035\u20131061.","DOI":"10.1214\/009053604000000283"},{"key":"20","doi-asserted-by":"publisher","unstructured":"[20] Glueck, D. H., Mandel, J., Karimpour-Fard, A., Hunter, L. and Muller, K. E. (2008). Exact calculations of average power for the Benjamini\u2013Hochberg procedure. <i>Int. J. Biostat.<\/i> <b>4<\/b> 1103.","DOI":"10.2202\/1557-4679.1103"},{"key":"21","doi-asserted-by":"publisher","unstructured":"[21] Guo, W. and Rao, M. B. (2008). On control of the false discovery rate under no assumption of dependency. <i>J. Statist. Plann. Inference<\/i> <b>138<\/b> 3176\u20133188.","DOI":"10.1016\/j.jspi.2008.01.003"},{"key":"22","doi-asserted-by":"publisher","unstructured":"[22] Lehmann, E. L. and Romano, J. P. (2005). Generalizations of the familywise error rate. <i>Ann. Statist.<\/i> <b>33<\/b> 1138\u20131154.","DOI":"10.1214\/009053605000000084"},{"key":"23","doi-asserted-by":"publisher","unstructured":"[23] Neuvial, P. (2008). Asymptotic properties of false discovery rate controlling procedures under independence. <i>Electron. J. Stat.<\/i> <b>2<\/b> 1065\u20131110.","DOI":"10.1214\/08-EJS207"},{"key":"24","doi-asserted-by":"publisher","unstructured":"[24] Owen, D. B. and Steck, G. P. (1962). Moments of order statistics from the equicorrelated multivariate normal distribution. <i>Ann. Math. Statist.<\/i> <b>33<\/b> 1286\u20131291.","DOI":"10.1214\/aoms\/1177704361"},{"key":"25","doi-asserted-by":"publisher","unstructured":"[25] Reiner-Benaim, A. (2007). FDR control by the BH procedure for two-sided correlated tests with implications to gene expression data analysis. <i>Biom. J.<\/i> <b>49<\/b> 107\u2013126.","DOI":"10.1002\/bimj.200510313"},{"key":"27","doi-asserted-by":"publisher","unstructured":"[27] Roquain, E. and van de Wiel, M. (2009). Optimal weighting for false discovery rate control. <i>Electron. J. Stat.<\/i> <b>3<\/b> 678\u2013711.","DOI":"10.1214\/09-EJS430"},{"key":"28","unstructured":"[28] Roquain, E. and Villers, F. (2010). Supplement to \u201cExact calculations for false discovery proportion with application to least favorable configurations.\u201d DOI: <a href=\"10.1214\/10-AOS847SUPP\">10.1214\/10-AOS847SUPP<\/a>."},{"key":"29","doi-asserted-by":"publisher","unstructured":"[29] Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. <i>Ann. Statist.<\/i> <b>30<\/b> 239\u2013257.","DOI":"10.1214\/aos\/1015362192"},{"key":"30","unstructured":"[30] Sarkar, S. K. (2008). On methods controlling the false discovery rate. <i>Sankhy\u0101<\/i> <b>70<\/b> 135\u2013168."},{"key":"31","doi-asserted-by":"crossref","unstructured":"[31] Seeger, P. (1968). A note on a method for the analysis of significances en masse. <i>Technometrics<\/i> <b>10<\/b> 586\u2013593.","DOI":"10.1080\/00401706.1968.10490605"},{"key":"33","doi-asserted-by":"publisher","unstructured":"[33] Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. <i>Biometrika<\/i> <b>73<\/b> 751\u2013754.","DOI":"10.1093\/biomet\/73.3.751"},{"key":"34","doi-asserted-by":"publisher","unstructured":"[34] Storey, J. D. (2003). The positive false discovery rate: A Bayesian interpretation and the <i>q<\/i>-value. <i>Ann. Statist.<\/i> <b>31<\/b> 2013\u20132035.","DOI":"10.1214\/aos\/1074290335"},{"key":"35","doi-asserted-by":"crossref","unstructured":"[35] Stuart, A. (1958). Equally correlated variates and the multinormal integral. <i>J. Roy. Statist. Soc. Ser. B<\/i> <b>20<\/b> 373\u2013378.","DOI":"10.1111\/j.2517-6161.1958.tb00301.x"},{"key":"36","doi-asserted-by":"publisher","unstructured":"[36] Zeisel, A., Zuk, O. and Domany, E. (2011). FDR control with adaptive procedures and FDR monotonicity. <i>Ann. Appl. Statist.<\/i> To appear. Available at <a href=\"ArXiv:0909.3704v1\">ArXiv:0909.3704v1<\/a>.","DOI":"10.1214\/10-AOAS399"},{"key":"8","unstructured":"[8] Dickhaus, T. (2008). False discovery rate and asymptotics. Ph.D. thesis, Heinrich-Heine Univ., D\u00fcsseldorf."},{"key":"26","unstructured":"[26] Roquain, E. (2007). Exceptional motifs in heterogeneous sequences. Contributions to theory and methodology of multiple testing. Ph.D. thesis, Univ. Paris XI."},{"key":"32","unstructured":"[32] Shorack, G. R. and Wellner, J. A. (1986). <i>Empirical Processes With Applications to Statistics<\/i>. Wiley, New York."}],"container-title":["The Annals of Statistics"],"original-title":[],"link":[{"URL":"https:\/\/projecteuclid.org\/download\/pdfview_1\/euclid.aos\/1297779857","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T21:23:31Z","timestamp":1791581011000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annals-of-statistics\/volume-39\/issue-1\/Exact-calculations-for-false-discovery-proportion-with-application-to-least\/10.1214\/10-AOS847.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,1]]},"references-count":36,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,2,1]]}},"URL":"https:\/\/doi.org\/10.1214\/10-aos847","relation":{},"ISSN":["0090-5364","2168-8966"],"issn-type":[{"value":"0090-5364","type":"print"},{"value":"2168-8966","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,2,1]]}}}