Sequential Nonparametric Testing with the Law of the Iterated Logarithm

• DOI: 10.48550/arxiv.1506.03486
• Publication Date: 2015-01-01
• Authors: Balsubramani, Akshay; Ramdas, Aaditya
• Keywords: FOS: Computer and information sciences; Computer Science - Machine Learning; Mathematics - Statistics Theory; Machine Learning (stat.ML); Statistics Theory (math.ST); 01 natural sciences; Machine Learning (cs.LG); Methodology (stat.ME); Statistics - Machine Learning; FOS: Mathematics; 0101 mathematics; Statistics - Methodology

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.