Sequential Nonparametric Testing with the Law of the Iterated Logarithm
This provides an efficient, adaptive solution for sequential testing problems like A/B testing, though it is incremental in applying LIL to nonparametric contexts.
The paper tackles sequential hypothesis testing for i.i.d. data, including A/B and nonparametric tests, by proposing a framework that runs in linear time with constant space, matches non-sequential test power up to a small factor, and adapts sample usage to problem difficulty, with simulations verifying error and stopping time predictions.
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 as a non-sequential version of the test with the same computational constraints up to a small factor, 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 the case of nonparametric two-sample mean testing, we also provide a finite sample power analysis, and the first non-asymptotic stopping time calculations for this class of problems. We verify our predictions for type I and II errors and stopping times using simulations.