The Annals of Statistics

Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure

Arthur Cohen and Harold B. Sackrowitz

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The problem of multiple endpoint testing for k endpoints is treated as a 2k finite action problem. The loss function chosen is a vector loss function consisting of two components. The two components lead to a vector risk. One component of the vector risk is the false rejection rate (FRR), that is, the expected number of false rejections. The other component is the false acceptance rate (FAR), that is, the expected number of acceptances for which the corresponding null hypothesis is false. This loss function is more stringent than the positive linear combination loss function of Lehmann [Ann. Math. Statist. 28 (1957) 1–25] and Cohen and Sackrowitz [Ann. Statist. (2005) 33 126–144] in the sense that the class of admissible rules is larger for this vector risk formulation than for the linear combination risk function. In other words, fewer procedures are inadmissible for the vector risk formulation. The statistical model assumed is that the vector of variables Z is multivariate normal with mean vector μ and known intraclass covariance matrix Σ. The endpoint hypotheses are Hii=0 vs Kii>0, i=1,…,k. A characterization of all symmetric Bayes procedures and their limits is obtained. The characterization leads to a complete class theorem. The complete class theorem is used to provide a useful necessary condition for admissibility of a procedure. The main result is that the step-up multiple endpoint procedure is shown to be inadmissible.

Article information

Ann. Statist., Volume 33, Number 1 (2005), 145-158.

First available in Project Euclid: 8 April 2005

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Zentralblatt MATH identifier

Primary: 62C10: Bayesian problems; characterization of Bayes procedures 62C15: Admissibility

Step-up procedure intraclass correlation Bayes procedures inadmissibility finite action problem Schur convexity complete class


Cohen, Arthur; Sackrowitz, Harold B. Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure. Ann. Statist. 33 (2005), no. 1, 145--158. doi:10.1214/009053604000000986.

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  • 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.
  • Brown, L. D., Cohen, A. and Strawderman, W. E. (1976). A complete class theorem for strict monotone likelihood ratio with applications. Ann. Statist. 4 712–722.
  • Brown, L. D., Johnstone, I. M. and MacGibbon, K. B. (1981). Variation diminishing transformation: A direct approach to total positivity and its statistical applications. J. Amer. Statist. Assoc. 76 824–832.
  • Cohen, A. and Sackrowitz, H. B. (1984). Decision theory results for vector risks with applications. Statist. Decisions Suppl. 1 159–176.
  • Cohen, A. and Sackrowitz, H. B. (2005). Decision theory results for one-sided multiple comparison procedures. Ann. Statist. 33 126–144.
  • Dudoit, S., Shaffer, J. P. and Boldrick, J. C. (2003). Multiple hypothesis testing in microarray experiments. Statist. Sci. 18 71–103.
  • Ferguson, T. S. (1967). Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York.
  • Hochberg, Y. (1988). A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75 800–802.
  • Hochberg, Y. and Tamhane, A. C. (1987). Multiple Comparison Procedures. Wiley, New York.
  • Karlin, S. and Rubin, H. (1956). The theory of decision procedures for distributions with monotone likelihood ratio. Ann. Math. Statist. 27 272–299.
  • Lehmann, E. L. (1957). A theory of some multiple decision problems. I. Ann. Math. Statist. 28 1–25.
  • Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
  • Matthes, T. K. and Truax, D. R. (1967). Tests of composite hypotheses for the multivariate exponential family. Ann. Math. Statist. 38 681–697.
  • Sarkar, S. (2002). Some results on false discovery rate in stepwise multiple testing procedures. Ann. Statist. 30 239–257.
  • Shaffer, J. P. (1995). Multiple hypothesis testing. Annual Review of Psychology 46 561–584.
  • Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 197–206. Univ. California Press, Berkeley.
  • Van Houwelingen, H. C. and Verbeek, A. (1985). On the construction of monotone symmetric decision rules for distributions with monotone likelihood ratio. Scand. J. Statist. 12 73–81.
  • Weiss, L. (1961). Statistical Decision Theory. McGraw-Hill, New York.