  A central problem in Binary Hypothesis Testing (BHT) is to determine the
optimal tradeoff between the Type I error (referred to as false alarm) and Type
II (referred to as miss) error. In this context, the exponential rate of
convergence of the optimal miss error probability -- as the sample size tends
to infinity -- given some (positive) restrictions on the false alarm
probabilities is a fundamental question to address in theory. Considering the
more realistic context of a BHT with a finite number of observations, this
paper presents a new non-asymptotic result for the scenario with monotonic
(sub-exponential decreasing) restriction on the Type I error probability, which
extends the result presented by Strassen in 2009. Building on the use of
concentration inequalities, we offer new upper and lower bounds to the optimal
Type II error probability for the case of finite observations. Finally, the
derived bounds are evaluated and interpreted numerically (as a function of the
number samples) for some vanishing Type I error restrictions.
