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In [[statistical hypothesis testing]], the error-exponent of a hypothesis testing procedure is the rate at which the error probability of a test decays exponentially with the number of samples used in the test. For example, if the probability of error <math>P_{\mathrm{error}}</math> of a test decays as <math>e^{-n \beta}</math>, where <math>n</math> is the sample size, the error exponent is <math>\beta</math>.
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: <math>\lim_{n \to \infty}\frac{- \ln P_{\mathrm{error}}}{n}</math>. Error-exponents for different hypothesis tests are computed using results from [[large deviations theory]].
==Error-exponents in binary hypothesis testing==
Consider a binary hypothesis testing problem in which observations are modeled as [[independent and identically distributed random variables]] under each hypothesis. Let <math> Y_1, Y_2, \ldots, Y_n </math> denote the observations. Let <math> f_0 </math> denote the probability density function of each observation <math>Y_i</math> under the null hypothesis <math>H_0</math> and let <math> f_1 </math> denote the probability density function of each observation <math>Y_i</math> under the alternate hypothesis <math>H_1</math>.
In this case there are two possible error events. Error of type 1 occurs when the null hypothesis is true and it is wrongly rejected. Error of type 2 occurs when the alternate hypothesis is true and null hypothesis is not rejected. The probability of type 1 error is denoted <math>P (\mathrm{error}|H_0)</math> and the probability of type 2 error is denoted <math>P (\mathrm{error}|H_1)</math>.
===Optimal error-exponent for Neyman-Pearson testing===
In the Neyman-Pearson<ref name="NeymanPearson1933"></ref> version of binary hypothesis testing, one is interested in minimizing the probability of type 2 error <math>P (\mathrm{error}|H_1)</math> subject to the constraint that the the probability of type 1 error <math>P (\mathrm{error}|H_0)</math> is less than or equal to a pre-specified level <math>\alpha</math>. In this setting, the optimal testing procedure is a [[likelihood-ratio test]]<ref name=LR></ref>. Furthermore, the optimal test guarantees that the type 2 error probability decays exponentially in the sample size <math>n</math> according to <math>\lim_{n \to \infty} \frac{- \ln P (\mathrm{error}|H_1)}{n} = D(f_0\|f_1)</math><ref name=CT></ref>. The error-exponent <math>D(f_0\|f_1)</math> 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.
===Optimal error-exponent for average error probability in Bayesian hypothesis testing===
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 <math> \pi_0 </math> denote the prior probability of hypothesis <math> H_0 </math>. In this case the average error probability is given by <math> P_{\mathrm{ave}} = \pi_0 P (\mathrm{error}|H_0) + (1-\pi_0)P (\mathrm{error}|H_1)</math>. In this setting again a likelihood ratio test<ref name=Poor></ref> is optimal and the optimal error decays as <math> \lim_{n \to \infty} \frac{- \ln P_{\mathrm{ave}} }{n} = C(f_0,f_1)</math> where <math>C(f_0,f_1)</math> represents the Chernoff-information between the two distributions defined as
<math> C(f_0,f_1) = \min_{\lambda \in [0,1]} \int (f_0(x))^\lambda (f_1(x))^{(1-\lambda)} dx </math>.
==References==
[[Category:Statistical hypothesis testing| ]]
[[Category:Information theory| ]]
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: <math>\lim_{n \to \infty}\frac{- \ln P_{\mathrm{error}}}{n}</math>. Error-exponents for different hypothesis tests are computed using results from [[large deviations theory]].
==Error-exponents in binary hypothesis testing==
Consider a binary hypothesis testing problem in which observations are modeled as [[independent and identically distributed random variables]] under each hypothesis. Let <math> Y_1, Y_2, \ldots, Y_n </math> denote the observations. Let <math> f_0 </math> denote the probability density function of each observation <math>Y_i</math> under the null hypothesis <math>H_0</math> and let <math> f_1 </math> denote the probability density function of each observation <math>Y_i</math> under the alternate hypothesis <math>H_1</math>.
In this case there are two possible error events. Error of type 1 occurs when the null hypothesis is true and it is wrongly rejected. Error of type 2 occurs when the alternate hypothesis is true and null hypothesis is not rejected. The probability of type 1 error is denoted <math>P (\mathrm{error}|H_0)</math> and the probability of type 2 error is denoted <math>P (\mathrm{error}|H_1)</math>.
===Optimal error-exponent for Neyman-Pearson testing===
In the Neyman-Pearson<ref name="NeymanPearson1933"></ref> version of binary hypothesis testing, one is interested in minimizing the probability of type 2 error <math>P (\mathrm{error}|H_1)</math> subject to the constraint that the the probability of type 1 error <math>P (\mathrm{error}|H_0)</math> is less than or equal to a pre-specified level <math>\alpha</math>. In this setting, the optimal testing procedure is a [[likelihood-ratio test]]<ref name=LR></ref>. Furthermore, the optimal test guarantees that the type 2 error probability decays exponentially in the sample size <math>n</math> according to <math>\lim_{n \to \infty} \frac{- \ln P (\mathrm{error}|H_1)}{n} = D(f_0\|f_1)</math><ref name=CT></ref>. The error-exponent <math>D(f_0\|f_1)</math> 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.
===Optimal error-exponent for average error probability in Bayesian hypothesis testing===
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 <math> \pi_0 </math> denote the prior probability of hypothesis <math> H_0 </math>. In this case the average error probability is given by <math> P_{\mathrm{ave}} = \pi_0 P (\mathrm{error}|H_0) + (1-\pi_0)P (\mathrm{error}|H_1)</math>. In this setting again a likelihood ratio test<ref name=Poor></ref> is optimal and the optimal error decays as <math> \lim_{n \to \infty} \frac{- \ln P_{\mathrm{ave}} }{n} = C(f_0,f_1)</math> where <math>C(f_0,f_1)</math> represents the Chernoff-information between the two distributions defined as
<math> C(f_0,f_1) = \min_{\lambda \in [0,1]} \int (f_0(x))^\lambda (f_1(x))^{(1-\lambda)} dx </math>.
==References==
[[Category:Statistical hypothesis testing| ]]
[[Category:Information theory| ]]
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