  The problem of binary hypothesis testing between two probability measures is
considered. New sharp bounds are derived for the best achievable error
probability of such tests based on independent and identically distributed
observations. Specifically, the asymmetric version of the problem is examined,
where different requirements are placed on the two error probabilities.
Accurate nonasymptotic expansions with explicit constants are obtained for the
error probability, using tools from large deviations and Gaussian
approximation. Examples are shown indicating that, in the asymmetric regime,
the approximations suggested by the new bounds are significantly more accurate
than the approximations provided by either of the two main earlier approaches
-- normal approximation and error exponents.
