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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.
 
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.
  
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Assume that the hypothesis $  H _ {0} $:  
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Assume that the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s0874801.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s0874802.png" /> has to be tested against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s0874803.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s0874804.png" /> by means of the realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s0874805.png" /> of a random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s0874806.png" /> that takes values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s0874807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s0874808.png" />. Furthermore, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s0874809.png" /> be an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748010.png" />-measurable function, mapping the sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748011.png" /> onto the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748012.png" />. In a case like this, the principle according to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748013.png" /> is rejected with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748014.png" />, while the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748015.png" /> is rejected with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748016.png" />, is called a statistical test for testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748017.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748018.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748019.png" /> is the critical function of the test. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748021.png" />, is called the power function of the test.
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$  \theta \in \Theta _ {0} \subset  \Theta $
 
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has to be tested against the alternative $  H _ {1} $:  
 
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$  \theta \in \Theta _ {1} = \Theta \setminus  \Theta _ {0} $
 
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by means of the realization $  x = ( x _ {1} \dots x _ {n} ) $
 
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of a random vector $  X = ( X _ {1} \dots X _ {n} ) $
 
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that takes values in a sample space $  ( \mathfrak X _ {n} , {\mathcal B} _ {n} , {\mathsf P} _  \theta  ^ {n} ) $,
 
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$  \theta \in \Theta $.  
 
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Furthermore, let $  \phi _ {n} ( \cdot ) $
 
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be an arbitrary $  {\mathcal B} _ {n} $-
 
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measurable function, mapping the sample space $  \mathfrak X _ {n} $
 
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onto the interval $  [ 0, 1] $.  
 
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In a case like this, the principle according to which $  H _ {0} $
 
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is rejected with probability $  \phi _ {n} ( X) $,  
 
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while the alternative $  H _ {1} $
 
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is rejected with probability $  1 - \phi _ {n} ( X) $,  
 
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is called a statistical test for testing $  H _ {0} $
 
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against $  H _ {1} $;  
 
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$  \phi _ {n} ( \cdot ) $
 
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is the critical function of the test. The function $  \beta ( \theta ) = {\mathsf E} _  \theta  \phi _ {n} ( X) $,  
 
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$  \theta \in \Theta $,  
 
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is called the power function of the test.
 
  
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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} $,  
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748022.png" />, and thus acceptance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748023.png" />, when in fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748024.png" /> is correct (an error of the first kind), or acceptance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748025.png" /> when in fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748026.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748028.png" />, for the probability of an error of the first kind, would minimize the probability of an error of the second kind. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748029.png" /> is called the [[Significance level|significance level]] of the statistical test.
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and thus acceptance of $  H _ {1} $,  
 
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when in fact $  H _ {0} $
 
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is correct (an error of the first kind), or acceptance of $  H _ {0} $
 
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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 ) $,  
 
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$  0 < \alpha < 1 $,  
 
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for the probability of an error of the first kind, would minimize the probability of an error of the second kind. The number $  \alpha $
 
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is called the [[Significance level|significance level]] of the statistical test.
 
  
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In practice, the most important are non-randomized statistical tests, i.e. those with as critical function $  \phi _ {n} ( \cdot ) $
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In practice, the most important are non-randomized statistical tests, i.e. those with as critical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748030.png" /> the indicator function of a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748031.png" />-measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748033.png" />:
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the indicator function of a certain $  {\mathcal B} _ {n} $-
 
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measurable set $  K $
 
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in $  \mathfrak X $:
 
  
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$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748034.png" /></td> </tr></table>
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\phi _ {n} ( x)  = \left \{
 
  
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Thus, a non-randomized statistical test rejects $  H _ {0} $
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Thus, a non-randomized statistical test rejects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748035.png" /> if the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748036.png" /> takes place; on the other hand, if the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748037.png" /> takes place, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748038.png" /> is accepted. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748039.png" /> is called the critical region of the statistical test.
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if the event $  \{ X \in K \} $
 
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takes place; on the other hand, if the event $  \{ X \in \overline{K}\; \} $
 
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takes place, then $  H _ {0} $
 
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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) $,  
+
As a rule, a non-randomized statistical test is based on a certain statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748040.png" />, which is called the test statistic, and the critical region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748041.png" /> of this same test is usually defined using relations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748044.png" />. The constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748046.png" />, called the critical values of the test statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748047.png" />, are defined from the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748048.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748049.png" /> reflects the particular nature of the competing hypotheses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748051.png" />. In the case where the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748052.png" /> possesses a sufficient statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748053.png" />, it is natural to look for the test statistic in the class of sufficient statistics, since
−
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} } \} $,  
 
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$  K = \{ {x } : {T _ {n} ( x) < t _ {1} } \} \cup \{ {x } : {T _ {n} ( x) > t _ {2} } \} $.  
 
−
The constants $  t _ {1} $,  
 
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$  t _ {2} $,  
 
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called the critical values of the test statistic $  T _ {n} $,  
 
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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} $.  
 
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In the case where the family $  \{ { {\mathsf P} _  \theta  ^ {n} } : {\theta \in \Theta } \} $
 
−
possesses a sufficient statistic $  \Psi = \Psi ( X) $,  
 
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it is natural to look for the test statistic in the class of sufficient statistics, since
 
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748054.png" /></td> </tr></table>
−
\beta _ {n} ( \theta )  = {\mathsf E} _  \theta  \phi _ {n} ( X)  = {\mathsf E} _  \theta  T _ {n} ( X)
 
−
$$
 
  
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for all $  \theta \in \Theta = \Theta _ {0} \cup \Theta _ {1} $,  
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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087480/s08748056.png" />.
−
where $  T _ {n} ( X) = {\mathsf E} \{ \phi _ {n} ( X)  \mid  \Psi \} $.
 
  
 
====References====
 
====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>
 
<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>

Revision as of 14:53, 7 June 2020

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

Assume that the hypothesis : has to be tested against the alternative : by means of the realization of a random vector that takes values in a sample space , . Furthermore, let be an arbitrary -measurable function, mapping the sample space onto the interval . In a case like this, the principle according to which is rejected with probability , while the alternative is rejected with probability , is called a statistical test for testing against ; is the critical function of the test. The function , , 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 , and thus acceptance of , when in fact is correct (an error of the first kind), or acceptance of when in fact 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 , , for the probability of an error of the first kind, would minimize the probability of an error of the second kind. The number is called the significance level of the statistical test.

In practice, the most important are non-randomized statistical tests, i.e. those with as critical function the indicator function of a certain -measurable set in :

Thus, a non-randomized statistical test rejects if the event takes place; on the other hand, if the event takes place, then is accepted. The set is called the critical region of the statistical test.

As a rule, a non-randomized statistical test is based on a certain statistic , which is called the test statistic, and the critical region of this same test is usually defined using relations of the form , , . The constants , , called the critical values of the test statistic , are defined from the condition ; 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 reflects the particular nature of the competing hypotheses and . In the case where the family possesses a sufficient statistic , it is natural to look for the test statistic in the class of sufficient statistics, since

for all , where .

References

[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2] J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)
[3] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[4] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
[5] 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)
[6] 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
How to Cite This Entry:
Statistical test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Statistical_test&oldid=49445
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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