Online multiple testing with e-values

Abstract: A scientist tests a continuous stream of hypotheses over time in the course of her investigation -- she does not test a predetermined, fixed number of hypotheses. The scientist wishes to make as many discoveries as possible while ensuring the number of false discoveries is controlled -- a well recognized way for accomplishing this is to control the false discovery rate (FDR). Prior methods for FDR control in the online setting have focused on formulating algorithms when specific dependency structures are assumed to exist between the test statistics of each hypothesis. However, in practice, these dependencies often cannot be known beforehand or tested after the fact. Our algorithm, e-LOND, provides FDR control under arbitrary, possibly unknown, dependence. We show that our method is more powerful than existing approaches to this problem through simulations. We also formulate extensions of this algorithm to utilize randomization for increased power, and for constructing confidence intervals in online selective inference.