# 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) |

**How to Cite This Entry:**

Critical function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Critical_function&oldid=46554