Testing Properties of Multiple Distributions with Few Samples

We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from s distributions, p1, p2, ldots, ps, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the pi's are uniform or ε-far from being uniform in ℓ1-distance (2) Identity Testing: Testing whether all the pi's are equal to an explicitly given distribution q or ε-far from q in ℓ1-distance, and (3) Closeness Testing: Testing whether all the pi's are equal to a distribution q which we have sample access to, or ε-far from q in ℓ1-distance. By assuming an additional natural condition about the source distributions, we provide sample optimal testers for all of these problems.