Critical function

A statistic for which the values are the conditional probabilities of the deviations from the hypothesis being tested, given the value of an observed result. Let $ X $ be a random variable with values in a sample space $ ( \mathfrak X , \mathfrak B ) $, the distribution of which belongs to a family $ \{ {P _ \theta } : {\theta \in \Theta } \} $, and suppose one is testing the hypothesis $ H _ {0} $: $ \theta \in \Theta _ {0} \subset \Theta $, against the alternative $ H _ {1} $: $ \theta \in \Theta _ {1} = \Theta \setminus \Theta _ {0} $. Let $ \phi ( \cdot ) $ be a measurable function on $ \mathfrak X $ such that $ 0 \leq \phi ( x) \leq 1 $ for all $ x \in \mathfrak X $. If the hypothesis is being tested by a randomized test, according to which $ H _ {0} $ is rejected with probability $ \phi ( x) $ if the experiment reveals that $ X = x $, and accepted with probability $ 1 - \phi ( x) $, then $ \phi ( \cdot ) $ is called the critical function of the test. In setting up a non-randomized test, one chooses the critical function in such a way that it assumes only two values, 0 and 1. Hence it is the characteristic function of a certain set $ K \in \mathfrak B $, called the critical region of the test: $ \phi ( x) = 1 $ if $ x \in K $, $ \phi ( x) = 0 $ if $ x \notin K $.

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)