  The problem of robust binary hypothesis testing is studied. Under both
hypotheses, the data-generating distributions are assumed to belong to
uncertainty sets constructed through moments; in particular, the sets contain
distributions whose moments are centered around the empirical moments obtained
from training samples. The goal is to design a test that performs well under
all distributions in the uncertainty sets, i.e., minimize the worst-case error
probability over the uncertainty sets. In the finite-alphabet case, the optimal
test is obtained. In the infinite-alphabet case, a tractable approximation to
the worst-case error is derived that converges to the optimal value using
finite samples from the alphabet. A test is further constructed to generalize
to the entire alphabet. An exponentially consistent test for testing batch
samples is also proposed. Numerical results are provided to demonstrate the
performance of the proposed robust tests.
