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 ${\displaystyle P_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}}$ 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: ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{-\ln P_{\text{error}}}{n}}$. Error exponents for different hypothesis tests are computed using Sanov's theorem and other results from large deviations theory. There are various methods used to show that an error exponent is achievable, including the likelihood ratio (which is known to be optimal in certain circumstances), and the empirical distribution^[1]. Error exponents are sometimes referred to as error rates, due to the connection between hypothesis testing and information theory^[2].

Consider a binary hypothesis testing problem in which observations are modeled as independent and identically distributed random variables under each hypothesis. Let ${\displaystyle Y_1,Y_2,\dots ,Y_n}$ denote the observations. Let ${\displaystyle f_0}$ denote the probability density function of each observation ${\displaystyle Y_i}$ under the null hypothesis ${\displaystyle H_0}$ and let ${\displaystyle f_1}$ denote the probability density function of each observation ${\displaystyle Y_i}$ under the alternate hypothesis ${\displaystyle H_1}$.

In this case there are two possible error events. Error of type I, also called false positive, occurs when the null hypothesis is true and it is wrongly rejected. Error of type II, also called false negative, occurs when the alternate hypothesis is true and null hypothesis is not rejected. The probability of type I error is denoted ${\displaystyle P(\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mid H_0)}$ and the probability of type II error is denoted ${\displaystyle P(\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mid H_1)}$. In some fields, the type I error is denoted by ${\displaystyle {\alpha }_n}$ and the type II error is denoted by ${\displaystyle {\beta }_n}$.

In the Neyman–Pearson^[3] version of binary hypothesis testing, one is interested in minimizing the probability of type II error ${\displaystyle P(\text{error}\mid H_1)}$ subject to the constraint that the probability of type I error ${\displaystyle P(\text{error}\mid H_0)}$ is less than or equal to a pre-specified level ${\displaystyle \alpha }$. In this setting, the optimal testing procedure is a likelihood-ratio test.^[4] Furthermore, the optimal test guarantees that the type II error probability decays exponentially in the sample size ${\displaystyle n}$ according to ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{-\ln P(\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mid H_1)}{n}=D(f_0\parallel f_1)}$.^[5] The error exponent ${\displaystyle D(f_0\parallel f_1)}$ is the Kullback–Leibler divergence between the probability distributions of the observations under the two hypotheses. This exponent is also referred to as the Chernoff–Stein lemma exponent.

In the Bayesian version of binary hypothesis testing one is interested in minimizing the average error probability under both hypothesis, assuming a prior probability of occurrence on each hypothesis. Let ${\displaystyle {\pi }_0}$ denote the prior probability of hypothesis ${\displaystyle H_0}$. In this case the average error probability is given by ${\displaystyle P_{\text{ave}}={\pi }_{0}P(\text{error}\mid H_0)+(1-{\pi }_0)P(\text{error}\mid H_1)}$. In this setting again a likelihood ratio test is optimal and the optimal error decays as ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{-\ln P_{\text{ave}}}{n}=C(f_0,f_1)}$ where ${\displaystyle C(f_0,f_1)}$ represents the Chernoff-information between the two distributions defined as ${\displaystyle C(f_0,f_1)=\underset{\lambda \in [0,1]}{\max }[-\ln \int (f_0(x){)}^{\lambda }(f_1(x){)}^{(1-\lambda )}dx]}$.^[5]

A more explicit tradeoff between the type I and type II error is observed when the type I error is constrained to decay exponentially, and the type II error is minimized. If we require

for some

, then the optimal type II error exponent is described by

. Here

is the Hoeffding divergence ^[6][7][2] described by

${\displaystyle H_r(f_0\parallel f_1)=\underset{0\le s\le 1}{\max }\frac{\mathrm{\Psi }(s)-(1-s)r}{s}}$ (1)

where

.

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Error exponents in hypothesis testing. Read more