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 $x \notin K$.