  We introduce a game-theoretic framework to study the hypothesis testing
problem, in the presence of an adversary aiming at preventing a correct
decision. Specifically, the paper considers a scenario in which an analyst has
to decide whether a test sequence has been drawn according to a probability
mass function (pmf) P_X or not. In turn, the goal of the adversary is to take a
sequence generated according to a different pmf and modify it in such a way to
induce a decision error. P_X is known only through one or more training
sequences. We derive the asymptotic equilibrium of the game under the
assumption that the analyst relies only on first order statistics of the test
sequence, and compute the asymptotic payoff of the game when the length of the
test sequence tends to infinity. We introduce the concept of
indistinguishability region, as the set of pmf's that can not be distinguished
reliably from P_X in the presence of attacks. Two different scenarios are
considered: in the first one the analyst and the adversary share the same
training sequence, in the second scenario, they rely on independent sequences.
The obtained results are compared to a version of the game in which the pmf P_X
is perfectly known to the analyst and the adversary.
