Neyman-Pearson Lemma

If there exists a critical region $C$ of size $\alpha$ and a Nonnegative constant $k$ such that

$${\prod_{i=1}^n f(x_i\vert\theta_1)\over \prod_{i=1}^n f(x_i\vert\theta_0)} \geq k$$

for points in $C$ and

$${\prod_{i=1}^n f(x_i\vert\theta_1)\over \prod_{i=1}^n f(x_i\vert\theta_0)} \leq k$$

for points not in $C$, then $C$ is a best critical region of size $\alpha$.

References

Hoel, P. G.; Port, S. C.; and Stone, C. J. ``Testing Hypotheses.'' Ch. 3 in Introduction to Statistical Theory. New York: Houghton Mifflin, pp. 56-67, 1971.