  In many real-world applications, it is common that a proportion of the data
may be missing or only partially observed. We develop a novel two-sample
testing method based on the Maximum Mean Discrepancy (MMD) which accounts for
missing data in both samples, without making assumptions about the missingness
mechanism. Our approach is based on deriving the mathematically precise bounds
of the MMD test statistic after accounting for all possible missing values. To
the best of our knowledge, it is the only two-sample testing method that is
guaranteed to control the Type I error for both univariate and multivariate
data where data may be arbitrarily missing. Simulation results show that our
method has good statistical power, typically for cases where 5% to 10% of the
data are missing. We highlight the value of our approach when the data are
missing not at random, a context in which either ignoring the missing values or
using common imputation methods may not control the Type I error.
