ASYMPTOTIC BAYES-OPTIMALITY UNDER SPARSITY OF SOME MULTIPLE TESTING PROCEDURES

被引:67
作者
Bogdan, Malgorzata [1 ,4 ]
Chakrabarti, Arijit [3 ]
Frommlet, Florian [2 ]
Ghosh, Jayanta K. [3 ,4 ]
机构
[1] Wroclaw Univ Technol, Inst Math & Comp Sci, PL-50370 Wroclaw, Poland
[2] Univ Vienna, Dept Stat & Decis Support Syst, A-1210 Vienna, Austria
[3] Indian Stat Inst, Kolkata 700108, W Bengal, India
[4] Purdue Univ, Dept Stat, W Lafayette, IN 47907 USA
关键词
Multiple testing; FDR; Bayes oracle; asymptotic optimality; FALSE DISCOVERY RATE; EMPIRICAL-BAYES; RATES; NULL; PROPORTION; HYPOTHESES; NUMBER;
D O I
10.1214/10-AOS869
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
Within a Bayesian decision theoretic framework we investigate some asymptotic optimality properties of a large class of multiple testing rules. A parametric setup is considered, in which observations come from a normal scale mixture model and the total loss is assumed to be the sum of losses for individual tests. Our model can be used for testing point null hypotheses, as well as to distinguish large signals from a multitude of very small effects. A rule is defined to be asymptotically Bayes optimal under sparsity (ABOS), if within our chosen asymptotic framework the ratio of its Bayes risk and that of the Bayes oracle (a rule which minimizes the Bayes risk) converges to one. Our main interest is in the asymptotic scheme where the proportion p of "true" alternatives converges to zero. We fully characterize the class of fixed threshold multiple testing rules which are ABOS, and hence derive conditions for the asymptotic optimality of rules controlling the Bayesian False Discovery Rate (BFDR). We finally provide conditions under which the popular Benjamini-Hochberg (BH) and Bonferroni procedures are ABOS and show that for a wide class of sparsity levels, the threshold of the former can be approximated by a nonrandom threshold. It turns out that while the choice of asymptotically optimal FDR levels for BH depends on the relative cost of a type I error, it is almost independent of the level of sparsity. Specifically, we show that when the number of tests in increases to infinity, then BH with FDR level chosen in accordance with the assumed loss function is ABOS in the entire range of sparsity parameters p proportional to m(-beta), with beta is an element of (0, 1].
引用
收藏
页码:1551 / 1579
页数:29
相关论文
共 44 条
[1]   Adapting to unknown sparsity by controlling the false discovery rate [J].
Abramovich, Felix ;
Benjamini, Yoav ;
Donoho, David L. ;
Johnstone, Iain M. .
ANNALS OF STATISTICS, 2006, 34 (02) :584-653
[2]   CONTROLLING THE FALSE DISCOVERY RATE - A PRACTICAL AND POWERFUL APPROACH TO MULTIPLE TESTING [J].
BENJAMINI, Y ;
HOCHBERG, Y .
JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY, 1995, 57 (01) :289-300
[3]  
BOGDAN M., 2010, ASYMPTOTIC BAYES O S, DOI [10.1214/10-AOS869SUPP, DOI 10.1214/10-AOS869SUPP]
[4]  
Bogdan M., 2008, IMS COLLECTIONS, V1, P211, DOI DOI 10.1214/193940307000000158
[5]  
BOGDAN M., 2010, BAYES ORACLE ASYMPTO
[6]   On the empirical Bayes approach to the problem of multiple testing [J].
Bogdan, Malgorzata ;
Ghosh, Jayanta K. ;
Ochman, Aleksandra ;
Tokdar, Surya T. .
QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL, 2007, 23 (06) :727-739
[7]   Estimation and confidence sets for sparse normal mixtures [J].
Cai, T. Tony ;
Jin, Jiashun ;
Low, Mark G. .
ANNALS OF STATISTICS, 2007, 35 (06) :2421-2449
[8]   OPTIMAL RATES OF CONVERGENCE FOR ESTIMATING THE NULL DENSITY AND PROPORTION OF NONNULL EFFECTS IN LARGE-SCALE MULTIPLE TESTING [J].
Cai, T. Tony ;
Jin, Jiashun .
ANNALS OF STATISTICS, 2010, 38 (01) :100-145
[9]  
CARVALHO C. M., 2008, 200831 DUKE U DEP ST
[10]   On the performance of FDR control: Constraints and a partial solution [J].
Chi, Zhiyi .
ANNALS OF STATISTICS, 2007, 35 (04) :1409-1431