Private and Non-private Uniformity Testing for Ranking Data

We study the problem of uniformity testing for statistical data that consists of rankings over m items, where the alternative class is restricted to Mallows models. Testing ranking data is challenging because of the size of the large domain that is factorial in m, therefore the tester needs to take advantage of some structure of the alternative class. We show that uniform distribution can be distinguished from Mallows model with O(m-(1/2)) samples based on simple pairwise statistics, which allows us to test uniformity using only two samples, if m is large enough. We also consider uniformity testing with central and local differential privacy (DP) constraints. We present a central DP algorithm that requires O(max{1/epsilon 0, 1/root m}), where epsilon(0) is the privacy budget parameter. Interestingly, our uniformity testing algorithm is straightforward to apply to the local DP scenario, since it works with binary statistics that is extracted from the ranking data. We carry out large-scale experiments, including m = 10, 000, to show that our uniformity testing algorithms scale gracefully with m.