Large-Scale Simultaneous Testing Using Kernel Density Estimation

A century ago, when Student’s t-statistic was introduced, no one ever imag-ined its increasing applicability in the modern era. It finds applications in highly multiple hypothesis testing, feature selection and ranking, high di-mensional signal detection, etc. Student’s t-statistic is constructed based on the empirical distribution function (EDF). An alternative choice to the EDF is the kernel density estimate (KDE), which is a smoothed version of the EDF. The novelty of the work consists of an alternative to Student’s t-test that uses the KDE technique and exploration of the usefulness of KDE based t-test in the context of its application to large-scale simultaneous hypothesis testing. An optimal bandwidth parameter for the KDE approach is derived by minimizing the asymptotic error between the true p-value and its asymptotic estimate based on normal approximation. If the KDE-based approach is used for large-scale simultaneous testing, then it is interesting to consider, when does the method fail to manage the error rate? We show that the suggested KDE-based method can control false discovery rate (FDR) if total number tests diverge at a smaller order of magnitude than N^3/2, where N is the total sample size. We compare our method to several possible al-ternatives with respect to FDR. We show in simulations that our method produces a lower proportion of false discoveries than its competitors. That is, our method better controls the false discovery rate than its competitors. Through these empirical studies, it is shown that the proposed method can be successfully applied in practice. The usefulness of the proposed methods is further illustrated through a gene expression data example.