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 be a random variable with values in a sample space
, the distribution of which belongs to a family
, and suppose one is testing the hypothesis
:
, against the alternative
:
. Let
be a measurable function on
such that
for all
. If the hypothesis is being tested by a randomized test, according to which
is rejected with probability
if the experiment reveals that
, and accepted with probability
, then
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
, called the critical region of the test:
if
,
if
.
References
[1] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959) |
Critical function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Critical_function&oldid=13465