Nonparametric inference for ratios of densities via uniformly valid and powerful permutation tests

Understand

We propose the density ratio permutation test, a hypothesis test that assesses whether the ratio between two densities is proportional to a known function based on independent samples from each distribution.

• The test uses an efficient Markov Chain Monte Carlo scheme to draw weighted permutations of the pooled data, yielding exchangeable samples and finite sample validity.
• For power, if the statistic is an integral probability metric, our procedure is consistent under mild assumptions on the defining function class; specializing to a reproducing kernel Hilbert space, we introduce the shifted maximum mean discrepancy and prove minimax optimality of our test when a normalized difference between the densities lies in a Sobolev ball.
• We extend to the case of an unknown density ratio by estimating it on an independent training sample and derive type~I error bounds in terms of the estimation error as well as power results.