The false discovery rate (FDR) introduced by Benjamini and Hochberg (1995) is perhaps the most standard error controlling measure being used in a wide variety of applications involving multiple hypothesis testing. There are two approaches to control the FDR - the fixed error rate approach of Benjamini and Hochberg (BH, 1995) where a rejection region is determined with the FDR below a fixed level and the estimation based approach of Storey (2002) where the FDR is estimated for a fixed rejection region before it is controlled. In this proposal, we concentrate on both these approaches and propose new, improved versions of some FDR controlling procedures available in the literature. A number of adaptive procedures have been put forward in the literature, each attempting to improve the method of Benjamini and Hochberg (1995), the BH method, by incorporating into this method an estimate of number true null hypotheses. Among these, the method of Benjamini, Krieger and Yekutieli (2006), the BKY method, has been receiving lots of attention recently. In this proposal, a variant of the BKY method is proposed by considering a different estimate of number true null hypotheses, which often outperforms the BKY method in terms of the FDR control and power. Storey's (2002) estimation based approach to controlling the FDR has been developed from a class of conservatively biased point estimates of the FDR under a mixture model for the underlying p-values and a fixed rejection threshold for each null hypothesis. An alternative class of point estimates of the FDR with uniformly smaller conservative bias is proposed under the same setup. Numerical evidence is provided to show that the mean squared error (MSE) is also often smaller for this new class of estimates. Compared to Storey's (2002), the present class provides a more powerful estimation based approach to controlling the FDR. / Statistics

DNA microarrays are a relatively new technology for assessing the expression levels of thousands of genes simultaneously. Researchers hope to find genes that are differentially expressed by hybridizing cDNA from known treatment sources with various genes spotted on the microarrays. The large number of tests involved in analyzing microarrays has raised new questions in multiple testing. Several approaches for identifying differentially expressed genes have been proposed. This paper considers two: (1) a mixed models approach, and (2) the Signiffcance Analysis of Microarrays.