Sequential Nonparametric Testing with the Law of the Iterated Logarithm

We propose a new algorithmic framework for sequential hypothesis testing with i.i.d. data, which includes A/B testing, nonparametric two-sample testing, and independence testing as special cases. It is novel in several ways: (a) it takes linear time and constant space to compute on the fly, (b) it has the same power guarantee (up to a small factor) as a nonsequential version of the test with the same computational constraints, and (c) it accesses only as many samples as are required – its stopping time adapts to the unknown difficulty of the problem. All our test statistics are constructed to be zero-mean martingales under the null hypothesis, and the rejection threshold is governed by a uniform non-asymptotic law of the iterated logarithm (LIL). For nonparametric two-sample mean testing, we also provide a finite-sample power analysis, and the first nonasymptotic stopping time analysis for this class of problems. We verify our predictions for type I and II errors and stopping times using simulations.