Multiple hypotheses testing is one of the most active research areas in statistics. The number of hypotheses can be finite or infinite. For a multiple hypothesis testing, an overall error criterion must be properly defined and different test procedures must be developed. In this thesis, we investigate situations of both finite and infinite hypotheses testing. Accordingly, the thesis will be roughly divided into two parts.
The first part of this thesis will focus on the finite hypotheses testing. We study the False Discovery Rate (FDR) proposed by Benjamini and Hochberg in 1995, as an error criterion for a multiple testing procedure. We first attempt to find a functional relationship between FDR and the more familiar family-wise error rate (FWER) in order to study the practical aspects of the two criteria and to get a controlling procedure of one from that of the other. A few new theoretic results are then presented about FDR and based on these results, a new and “suboptimal” FDR controlling procedure is proposed. Some comparisons are made to compare the performance of the proposed procedure with that of Benjamini and Hochberg’s (1995) and Storey et al’s (2003). The procedure is then applied to a microarray data set to illustrate its application in the bioinformatics area.
The second part of this thesis involves testing the equality of two curves. This type of testing problems occurs often in functional data analysis. In this part, we develop test procedures for testing if two curves measured with homoscedastic or heteroscedastic errors are equal. The method is applicable to a general class of curves that can be either specified up to some unknown parameters, or are only assumed to be smooth. The null distribution of the test statistic is derived and an approximation formula to estimate the p-value is developed, when the homoscedastic or heteroscedastic variances are either known or unknown. Simulation experiments are conducted to show how our procedures perform in finite sample situations. Application to our motivating data example from an environmental study is illustrated.
The two areas are actually related. We will discuss their connections in the last chapter and propose questions for future research.