Adaptive Binning Coincidence Test for Uniformity Testing

We consider the problem of uniformity testing of Lipschitz continuous distributions with bounded support. The alternative hypothesis is a composite set of Lipschitz continuous distributions whose
$\ell_{1}$
distances from the uniform distribution are bounded by
$\varepsilon$
from below. We propose a sequential test that adapts to the unknown distribution under the alternative hypothesis. Referred to as the Adaptive Binning Coincidence (ABC) test, the proposed strategy adapts in two ways. First, it partitions the set of alternative distributions into layers based on their distances to the uniform distribution. It then sequentially eliminates the alternative distributions layer by layer in decreasing distance to the uniform, allowing it to take advantage of favorable situations of a distant alternative by terminating early. Second, it adapts, across layers of the alternative distributions, the resolution level of the discretization for computing the coincidence statistic. The farther away the layer is from the uniform, the coarser the discretization necessary for eliminating this layer or terminating altogether. It thus terminates the test both
early
(via the layered partition of the alternative set) and
quickly
(via adaptive discretization) to take advantage of favorable alternative distributions. The ABC test builds on an adaptive sequential test for discrete distributions, which is of independent interest.