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A decision rule according to which a decision is taken in the problem of statistical hypotheses testing (cf. [[Statistical hypotheses, verification of|Statistical hypotheses, verification of]]) on the basis of results of observations.

Assume that the hypothesis  $  H _ {0} $: 
$  \theta \in \Theta _ {0} \subset  \Theta $
has to be tested against the alternative  $  H _ {1} $: 
$  \theta \in \Theta _ {1} = \Theta \setminus  \Theta _ {0} $
by means of the realization  $  x = ( x _ {1} \dots x _ {n} ) $
of a random vector  $  X = ( X _ {1} \dots X _ {n} ) $
that takes values in a sample space  $  ( \mathfrak X _ {n} , {\mathcal B} _ {n} , {\mathsf P} _  \theta   ^ {n} ) $, 
$  \theta \in \Theta $. 
Furthermore, let  $  \phi _ {n} ( \cdot ) $
be an arbitrary  $  {\mathcal B} _ {n} $-
measurable function, mapping the sample space  $  \mathfrak X _ {n} $
onto the interval  $  [ 0, 1] $. 
In a case like this, the principle according to which  $  H _ {0} $
is rejected with probability  $  \phi _ {n} ( X) $, 
while the alternative  $  H _ {1} $
is rejected with probability  $  1 - \phi _ {n} ( X) $, 
is called a statistical test for testing  $  H _ {0} $
against  $  H _ {1} $; 
$  \phi _ {n} ( \cdot ) $
is the critical function of the test. The function  $  \beta ( \theta ) = {\mathsf E} _  \theta  \phi _ {n} ( X) $, 
$  \theta \in \Theta $, 
is called the power function of the test.

The use of a statistical test leads either to a correct decision being taken, or to one of the following two errors being made: rejection of  $  H _ {0} $, 
and thus acceptance of  $  H _ {1} $, 
when in fact  $  H _ {0} $
is correct (an error of the first kind), or acceptance of  $  H _ {0} $
when in fact  $  H _ {1} $
is correct (an error of the second kind). One of the basic problems in the classical theory of statistical hypotheses testing is the construction of a test that, given a definite upper bound  $  \alpha = \sup _ {\theta \in \Theta _ {0}  }  \beta _ {n} ( \theta ) $, 
$  0 < \alpha < 1 $, 
for the probability of an error of the first kind, would minimize the probability of an error of the second kind. The number  $  \alpha $
is called the [[Significance level|significance level]] of the statistical test.

In practice, the most important are non-randomized statistical tests, i.e. those with as critical function  $  \phi _ {n} ( \cdot ) $
the indicator function of a certain  $  {\mathcal B} _ {n} $-
measurable set  $  K $
in  $  \mathfrak X $:

$$ 
\phi _ {n} ( x)  =  \left \{

\begin{array}{ll}
 1  & \textrm{ if }  x \in K,  \\
 0  & \textrm{ if }  x \in \overline{K}\; = \mathfrak X _ {n} \setminus  K.  \\
\end{array}

 \right .$$

Thus, a non-randomized statistical test rejects  $  H _ {0} $
if the event  $  \{ X \in K \} $
takes place; on the other hand, if the event  $  \{ X \in \overline{K}\; \} $
takes place, then  $  H _ {0} $
is accepted. The set  $  K $
is called the critical region of the statistical test.

As a rule, a non-randomized statistical test is based on a certain statistic  $  T _ {n} = T _ {n} ( X) $, 
which is called the test statistic, and the critical region  $  K $
of this same test is usually defined using relations of the form  $  K = \{ {x } : {T _ {n} ( x) < t _ {1} } \} $, 
$  K = \{ {x } : {T _ {n} ( x) > t _ {2} } \} $, 
$  K = \{ {x } : {T _ {n} ( x) < t _ {1} } \} \cup \{ {x } : {T _ {n} ( x) > t _ {2} } \} $. 
The constants  $  t _ {1} $, 
$  t _ {2} $, 
called the critical values of the test statistic  $  T _ {n} $, 
are defined from the condition  $  \alpha = \sup _ {\theta \in \Theta _ {0}  }  \beta _ {n} ( \theta ) $; 
in these circumstances one speaks in the first two cases of one-sided statistical tests, and in the third case, of a two-sided statistical test. The structure of  $  T _ {n} $
reflects the particular nature of the competing hypotheses  $  H _ {0} $
and  $  H _ {1} $. 
In the case where the family  $  \{ { {\mathsf P} _  \theta   ^ {n} } : {\theta \in \Theta } \} $
possesses a [[sufficient statistic]]  $  \Psi = \Psi ( X) $, 
it is natural to look for the test statistic in the class of sufficient statistics, since

$$ 
\beta _ {n} ( \theta )  =  {\mathsf E} _  \theta  \phi _ {n} ( X)  =  {\mathsf E} _  \theta  T _ {n} ( X)
$$

for all  $  \theta \in \Theta = \Theta _ {0} \cup \Theta _ {1} $, 
where  $  T _ {n} ( X) = {\mathsf E} \{ \phi _ {n} ( X)  \mid   \Psi \} $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Hájek,   Z. Sidák,   "Theory of rank tests" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.L. van der Waerden,   "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.N. Bol'shev,   N.V. Smirnov,   "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka  (1983)  (In Russian)  (Processed by L.S. Bark and E.S. Kedrova)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.S. Nikulin,   "A result of Bol'shev's from the theory of the statistical testing of hypotheses"  ''J. Soviet Math.'' , '''44''' :  3  (1989)  pp. 522–529  ''Zap. Nauchn. Sem. Mat. Inst. Steklov.'' , '''153'''  (1986)  pp. 129–137</TD></TR></table>