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Nonparametric two-sample tests such as the Maximum Mean Discrepancy (MMD) areoften used to detect differences between two distributions in machine learningapplications. However, the majority of existing literature assumes thaterror-free samples from the two distributions of interest are available.Werelax this assumption and study the estimation of the MMD under$\epsilon$-contamination, where a possibly non-random $\epsilon$ proportion ofone distribution is erroneously grouped with the other. We show that under$\epsilon$-contamination, the typical estimate of the MMD is unreliable.Instead, we study partial identification of the MMD, and characterize sharpupper and lower bounds that contain the true, unknown MMD. We propose a methodto estimate these bounds, and show that it gives estimates that converge to thesharpest possible bounds on the MMD as sample size increases, with aconvergence rate that is faster than alternative approaches. Using threedatasets, we empirically validate that our approach is superior to thealternatives: it gives tight bounds with a low false coverage rate.

Partial identification of kernel based two sample tests with mismeasured data