Abstract 

We study the general problem of testing whether an unknown discrete distribution belongs to a specified family of distributions. More specifically, given a distribution family P and sample access to an unknown discrete distribution P, we want to distinguish (with high probability) between the case that P P and the case that P is -far, in total variation distance, from every distribution in P. This is the prototypical hypothesis testing problem that has received significant attention in statistics and, more recently, in computer science. The main contribution of this work is a simple and general testing technique that is applicable to all distribution families whose Fourier spectrum satisfies a certain approximate sparsity property. We apply our Fourier-based framework to obtain near sample-optimal and computationally efficient testers for the following fundamental distribution families: Sums of Independent Integer Random Variables (SIIRVs), Poisson Multinomial Distributions (PMDs), and Discrete Log-Concave Distributions. For the first two, ours are the first non-trivial testers in the literature, vastly generalizing previous work on testing Poisson Binomial Distributions. For the third, our tester improves on prior work in both sample and time complexity.