On the false discovery rate and an asymptotically optimal rejection curve

In this paper we introduce and investigate a new rejection curve for asymptotic control of the false discovery rate (FDR) in multiple hypotheses testing problems. We first give a heuristic motivation for this new curve and propose some procedures related to it. Then we introduce a set of possible assumptions and give a unifying short proof of FDR control for procedures based on Simes' critical values, whereby certain types of dependency are allowed. This methodology of proof is then applied to other fixed rejection curves including the proposed new curve. Among others, we investigate the problem of finding least favorable parameter configurations such that the FDR becomes largest. We then derive a series of results concerning asymptotic FDR control for procedures based on the new curve and discuss several example procedures in more detail. A main result will be an asymptotic optimality statement for various procedures based on the new curve in the class of fixed rejection curves. Finally, we briefly discuss strict FDR control for a finite number of hypotheses.

[1]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[2]  R. Wijsman,et al.  A Useful Inequality on Ratios of Integrals, with Application to Maximum Likelihood Estimation , 1985 .

[3]  R. Simes,et al.  An improved Bonferroni procedure for multiple tests of significance , 1986 .

[4]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[5]  Anthony J. Hayter,et al.  On the Joint Distribution Functions of Order Statistics with Reference to Step-up Multiple Test Procedures , 1993, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[6]  Y. Benjamini,et al.  Controlling the false discovery rate: a practical and powerful approach to multiple testing , 1995 .

[7]  D. Donoho,et al.  Asymptotic Minimaxity Of False Discovery Rate Thresholding For Sparse Exponential Data , 2006, math/0602311.

[8]  A. Tamhane,et al.  A generalized step‐up‐down multiple test procedure , 1998 .

[9]  Y. Benjamini,et al.  THE CONTROL OF THE FALSE DISCOVERY RATE IN MULTIPLE TESTING UNDER DEPENDENCY , 2001 .

[10]  H. Finner,et al.  On the False Discovery Rate and Expected Type I Errors , 2001 .

[11]  S. Sarkar.  Some Results on False Discovery Rate in Stepwise multiple testing procedures , 2002 .

[12]  L. Wasserman,et al.  Operating characteristics and extensions of the false discovery rate procedure , 2002 .

[13]  John D. Storey,et al.  Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: a unified approach , 2004 .

[14]  D. Donoho,et al.  Higher criticism for detecting sparse heterogeneous mixtures , 2004, math/0410072.

[15]  L. Wasserman,et al.  A stochastic process approach to false discovery control , 2004, math/0406519.

[16]  I. Johnstone,et al.  Adapting to unknown sparsity by controlling the false discovery rate , 2005, math/0505374.

[17]  M. Kosorok,et al.  Marginal asymptotics for the “large $p$, small $n$” paradigm: With applications to microarray data , 2005, math/0508219.

[18]  S. Sarkar.  False discovery and false nondiscovery rates in single-step multiple testing procedures , 2006, math/0605607.

[19]  Y. Benjamini,et al.  Adaptive linear step-up procedures that control the false discovery rate , 2006 .

[20]  N. Meinshausen,et al.  Estimating the proportion of false null hypotheses among a large number of independently tested hypotheses , 2005, math/0501289.

[21]  B. Efron.  Size, power and false discovery rates , 2007, 0710.2245.

[22]  T. Dickhaus,et al.  Dependency and false discovery rate: Asymptotics , 2007, 0710.3171.

[23]  Zhiyi Chi.  On the performance of FDR control: Constraints and a partial solution , 2007, 0710.3287.

[24]  Y. Benjamini,et al.  An adaptive step-down procedure with proven FDR control under independence , 2009, 0903.5373.