  Uniformity testing and the more general identity testing are well studied
problems in distributional property testing. Most previous work focuses on
testing under $L_1$-distance. However, when the support is very large or even
continuous, testing under $L_1$-distance may require a huge (even infinite)
number of samples. Motivated by such issues, we consider the identity testing
in Wasserstein distance (a.k.a. transportation distance and earthmover
distance) on a metric space (discrete or continuous).
  In this paper, we propose the Wasserstein identity testing problem (Identity
Testing in Wasserstein distance). We obtain nearly optimal worst-case sample
complexity for the problem. Moreover, for a large class of probability
distributions satisfying the so-called "Doubling Condition", we provide nearly
instance-optimal sample complexity.
