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On the Simes inequality and its generalization

Sanat K. Sarkar

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The Simes inequality has received considerable attention recently because of its close connection to some important multiple hypothesis testing procedures. We revisit in this article an old result on this inequality to clarify and strengthen it and a recently proposed generalization of it to offer an alternative simpler proof.

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N. Balakrishnan, Edsel A. Peña and Mervyn J. Silvapulle, eds., Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 231-242

First available in Project Euclid: 1 April 2008

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Primary: 62G30: Order statistics; empirical distribution functions 62H15: Hypothesis testing

multivariate totally positive of order two positive dependence through stochastic ordering probability inequalities Simes test symmetric multivariate normal symmetric multivariate t

Copyright © 2008, Institute of Mathematical Statistics


Sarkar, Sanat K. On the Simes inequality and its generalization. Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, 231--242, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/193940307000000167.

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  • [1] 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.
  • [2] Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Ann. Statist. 29 1165–1188.
  • [3] Block, H. W., Savits, T. H. and Shaked, M. (1985). A concept of negative dependence using stochastic ordering. Statist. Probab. Lett. 3 81–86.
  • [4] Cai, G. and Sarkar, S. K. (2005). Modified Simes’ critical values under independence. Technical report, Temple Univ. Available at˜sanat/reports/Modified-Simes_Independence.pdf.
  • [5] Cai, G. and Sarkar, S. K. (2006). Modified Simes’ critical values under positive dependence. J. Statist. Plann. Inference 136 4129–4146.
  • [6] Deshpande, J. V. and Kochar, S. C. (1980). Some competitors of tests based on powers of ranks for the two-sample problem. Sankhyā Ser. B 42 236–241.
  • [7] Dmitrienko, A., Offen, W. and Westfall, P. (2003). Gatekeeping startegies for clinical trials that do not require all primary effects to be significant. Statist. Med. 22 2387–2400.
  • [8] Hochberg, Y. (1988). A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75 800–802.
  • [9] Hochberg, Y. and Liberman, U. (1994). An extended Simes’ test. Statist. Probab. Lett. 21 101–105.
  • [10] Hochberg, Y. and Rom, D. (1995). Extensions of multiple testing procedures based on Simes’ test. J. Statist. Plann. Inference 48 141–152.
  • [11] Hommel, G., Lindig, V. and Faldum, A. (2005). Two-stage adaptive designs with correlated test statistics. J. Biopharm. Statist. 15 613–623.
  • [12] 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.
  • [13] Karlin, S. and Rinott, Y. (1981). Total positivity properties of absolute value multinormal variables with applications to confidence interval estimates and related probabilistic inequalities. Ann. Statist. 9 1035–1049.
  • [14] Kochar, S. C. (1978). A class of distribution-free tests for the two-sample slippage problem. Comm. Statist. A Theory Methods 7 1243–1252.
  • [15] Krummenauer, F. and Hommel, G. (1999). The size of Simes’ global test for discrete test statistics. J. Statist. Plann. Inference 82 151–162.
  • [16] Meng, Z., Zaykin, D., Karnoub, M., Sreekumar, G., St Jean, P. and Ehm, M. (2001). Identifying susceptibility genes using linkage and linkage disequilibrium analysis in large pedigrees. Gen. Epidem. 21 S453–S458.
  • [17] Neuhauser, M., Steinijans, V. and Bretz, F. (1999). The evaluation of multiple clinical endpoints, with application to asthma. Drug Inf. J. 33 471–477.
  • [18] Rødland, E. A. (2006). Simes’ procedure is ‘valid on average’. Biometrika 93 742–746.
  • [19] Rosenberg, P. S., Che, A. and Chen, B. E. (2006). Multiple hypothesis testing strategies for genetic case-control association studies. Stat. Med. 25 3134–3149.
  • [20] Samuel-Cahn, E. (1996). Is the Simes improved Bonferroni procedure conservative? Biometrika 83 928–933.
  • [21] Samuel-Cahn, E. (1999). A note about a curious generalization of Simes’ theorem. J. Statist. Plann. Inference 82 147–149.
  • [22] Sarkar, S. K. (1998). Some probability inequalities for ordered MTP2 random variables: a proof of the Simes conjecture. Ann. Statist. 26 494–504.
  • [23] Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. Ann. Statist. 30 239–257.
  • [24] Sarkar, S. K. (2004). FDR-controlling stepwise procedures and their false negatives rates. J. Statist. Plann. Inference 125 119–137.
  • [25] Sarkar, S. K. (2006). False discovery and false nondiscovery rates in single-step multiple testing procedures. Ann. Statist. 34 394–415.
  • [26] Sarkar, S. K. (2007a). Stepup procedures controlling generalized FWER and generalized FDR. Ann. Statist. To appear.
  • [27] Sarkar, S. K. (2007b). Generalizing Simes’ test and Hochberg’s stepup procedure. Ann. Statist. To appear.
  • [28] Sarkar, S. K. (2007c). Two-stage stepup procedures controlling FDR. J. Statist. Plann. Inference. To appear.
  • [29] Sarkar, S. K. and Chang, C.-K. (1997). The Simes method for multiple hypothesis testing with positively dependent test statistics. J. Amer. Statist. Assoc. 92 1601–1608.
  • [30] Sarkar, S. K. and Guo, W. (2006). Procedures controlling generalized false discovery rate. Technical report, Temple Univ. Available at http://astro. ̃ sanat/reports/GeneralizedFDR.pdf.
  • [31] Sarkar, S. K. and Guo, W. (2007). On generalized false discovery rate. Unpublished manuscript.
  • [32] Sen, P. K. (1999). Some remarks on Simes-type multiple tests of significance. J. Statist. Plann. Inference 82 139–145.
  • [33] Sen, P. K. and Silvapulle, M. J. (2002). An appraisal of some aspects of statistical inference under inequality constraints. J. Statist. Plann. Inference 107 3–43.
  • [34] Seneta, E. and Chen, J. (2005). Simple stepwise tests of hypotheses and multiple comparisons. Internat. Statist. Rev. 73 21–34.
  • [35] Silvapulle, M. J. and Sen, P. K. (2004). Constrained Statistical Inference. Wiley, New York.
  • [36] Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. Biometrika 73 751–754.
  • [37] Somerville, M., Wilson, T., Koch, G. and Westfall, P. (2005). Evaluation of a weighted multiple comparison procedure. Pharm. Statist. 4 7–13.
  • [38] Westfall, P. H. and Krishen, A. (2001). Optimally weighted, fixed sequence and gatekeeper multiple testing procedures. J. Statist. Plann. Inference 99 25–40.