Wednesday, September 16, 2009
03:30 PM in REC 315
Professor Malgorzata Bogdan
Visiting Professor, Department of Statistics, Purdue University

On the asymptotic optimality of the Benjamini and Hochberg procedure

Abstract

We will continue the discussion on the asymptotic optimality of multiple testing rules within the framework of Bayesian Decision Theory. Similarly as in [1] our main interest is in the asymptotic scheme under which the proportion of "true" alternatives converges to zero as the number of tests increases to infinity. According to our definition the multiple testing rule is asymptotically optimal if the ratio of its Bayes risk and that of the Bayes oracle (a rule which minimizes the Bayes risk) converges to one within this asymptotic framework. We characterize the set of fixed threshold multiple testing rules which asymptotically optimal and based on this characterization discuss some important scenarios under which the Bonferroni correction and the fixed threshold rules controlling the Bayesian False Discovery Rate (BFDR) are asymptotically optimal. Finally, we present results on the convergence of the random threshold of the Benjamini and Hochberg procedure to the threshold of BFDR controlling rule and on the asymptotic optimality of the Benjamini and Hochberg procedure under a wide range of sparsity levels and relatively mild assumptions on the ratio of losses for type I and type II errors. As far as we know, this is the first result on the decision theoretic optimality of the Benjamini-Hochberg rule in the context of hypothesis testing.

References

  1. Abramovich F., Benjamini Y., Donoho D. L. and Johnstone I. M. (2006). Adapting to unknown sparsity by controlling the false discovery rate. Ann. Statist., 34, 584—653.
  2. Bogdan M., Chakrabarti A. and Ghosh J.K. (2009) Bayes Oracle and the Asymptotic Optimality of the Multiple Testing Procedures Under Sparsity, Tech Report 02/09 Purdue University.
  3. Bogdan M, Chakrabarti A., Frommlet F. and Ghosh J.K. (2009) On the Asymptotic Optimality of the Multiple Testing Rules Under Sparsity, in preparation.