Error exponents in hypothesis testing

In statistical hypothesis testing, the error exponent of a hypothesis testing procedure is the rate at which the probabilities of Type I and Type II decay exponentially with the size of the sample used in the test. For example, if the probability of error ${\displaystyle P_{\mathrm {error} }}$ of a test decays as ${\displaystyle e^{-n\beta }}$, where ${\displaystyle n}$ is the sample size, the error exponent is ${\displaystyle \beta }$.

Formally, the error exponent of a test is defined as the limiting value of the ratio of the negative logarithm of the error probability to the sample size for large sample sizes: ${\displaystyle \lim _{n\to \infty }{\frac {-\ln P_{\text{error}}}{n}}}$. Error exponents for different hypothesis tests are computed using Sanov's theorem and other results from large deviations theory.