Error exponents in hypothesis testing ($E={\lim }_{n\to \infty }-n^{-1}\log p_n$)

An error exponent is the limiting exponential decay rate of an error probability $p_n$ as sample size $n$ grows. Different constraints on the Type I error and Type II error give different optimal rates. The Chernoff information describes the optimal average-error rate for simple Bayesian hypotheses; Stein's lemma gives the optimal type-II rate when type-I error is held below a fixed nontrivial bound.