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A [[Non-parametric test|non-parametric test]] for a hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s0850401.png" />, according to which a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s0850402.png" /> has a binomial distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s0850403.png" />. If the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s0850404.png" /> is true, then
s0850401.png
 
$#A+1 = 25 n = 0
 
$#C+1 = 25 : ~/encyclopedia/old_files/data/S085/S.0805040 Sign test
 
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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/s085/s085040/s0850405.png" /></td> </tr></table>
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A [[Non-parametric test|non-parametric test]] for a hypothesis  $  H _ {0} $,
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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/s085/s085040/s0850406.png" /></td> </tr></table>
according to which a random variable  $  \mu $
 
has a binomial distribution with parameters  $  ( n ; p = 0 . 5 ) $.  
 
If the hypothesis  $  H _ {0} $
 
is true, then
 
 
 
$$
 
{\mathsf P} \left \{ \mu \leq  k \left | n ,
 
\frac{1}{2}
 
\right . \right \}  =  \sum _
 
{i = 0 } ^ { k }  \left ( \begin{array}{c}
 
n \\
 
i
 
\end{array}
 
\right ) \left (
 
\frac{1}{2}
 
\right )  ^ {n}  = \
 
I _ {0,5} ( n - k , k + 1 ) ,
 
$$
 
 
 
$$
 
k  =  0 \dots n ,
 
$$
 
  
 
where
 
where
  
$$
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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/s085/s085040/s0850407.png" /></td> </tr></table>
I _ {z} ( a , b )  =
 
\frac{1}{B ( a , b ) }
 
 
 
\int\limits _ { 0 } ^ { z }  t  ^ {a-} 1 ( 1 - t )  ^ {b-} 1 dt ,\ \
 
0 \leq  z \leq  1 ,
 
$$
 
  
and $  B ( a , b ) $
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and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s0850408.png" /> is the beta-function. According to the sign test with significance level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s0850409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504010.png" />, the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504011.png" /> is rejected if
is the beta-function. According to the sign test with significance level $  \alpha $,  
 
$  0 < \alpha \leq  0 . 5 $,  
 
the hypothesis $  H _ {0} $
 
is rejected if
 
  
$$
+
<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/s085/s085040/s08504012.png" /></td> </tr></table>
\min \{ \mu , n - \mu \}  \leq  m ,
 
$$
 
  
where $  m = m ( \alpha , n ) $,  
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504013.png" />, the critical value of the test, is the integer solution of the inequalities
the critical value of the test, is the integer solution of the inequalities
 
  
$$
+
<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/s085/s085040/s08504014.png" /></td> </tr></table>
\sum _ {i = 0 } ^ { m }  \left ( \begin{array}{c}
 
n \\
 
i
 
\end{array}
 
\right )
 
\left (
 
\frac{1}{2}
 
\right )  ^ {n}  \leq 
 
\frac \alpha {2}
 
,\ \
 
\sum _ {i = 0 } ^ { {m }  + 1 } \left ( \begin{array}{c}
 
n \\
 
i
 
\end{array}
 
\right )
 
\left (
 
\frac{1}{2}
 
\right )  ^ {n}  >
 
\frac \alpha {2}
 
.
 
$$
 
  
The sign test can be used to test a hypothesis $  H _ {0} $
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The sign test can be used to test a hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504015.png" /> according to which the unknown continuous distribution of independent identically-distributed random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504016.png" /> is symmetric about zero, i.e. for any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504017.png" />,
according to which the unknown continuous distribution of independent identically-distributed random variables $  X _ {1} \dots X _ {n} $
 
is symmetric about zero, i.e. for any real $  x $,
 
  
$$
+
<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/s085/s085040/s08504018.png" /></td> </tr></table>
{\mathsf P} \{ X _ {i} \langle  - x \}  = {\mathsf P} \{ X _ {i} \rangle x \} .
 
$$
 
  
 
In this case the sign test is based on the statistic
 
In this case the sign test is based on the statistic
  
$$
+
<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/s085/s085040/s08504019.png" /></td> </tr></table>
\mu  = \sum _ { i= } 1 ^ { n }  \delta ( X _ {i} ) ,\ \
 
\delta ( x)  = \left \{
 
  
which is governed by a binomial law with parameters $  ( n ;  p = 0 . 5 ) $
+
which is governed by a binomial law with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504020.png" /> if the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504021.png" /> is true.
if the hypothesis $  H _ {0} $
 
is true.
 
  
Similarly, the sign test is used to test a hypothesis $  H _ {0} $
+
Similarly, the sign test is used to test a hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504022.png" /> according to which the median of an unknown continuous distribution to which independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504023.png" /> are subject is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504024.png" />; to this end one simply replaces the given random variables by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s08504025.png" />.
according to which the median of an unknown continuous distribution to which independent random variables $  X _ {1} \dots X _ {n} $
 
are subject is $  \xi _ {0} $;  
 
to this end one simply replaces the given random variables by $  Y _ {1} = X _ {1} - \xi _ {0} \dots Y _ {n} = X _ {n} - \xi _ {0} $.
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.V. Smirnov,  I.V. Dunin-Barkovskii,  "Mathematische Statistik in der Technik" , Deutsch. Verlag Wissenschaft.  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.V. Smirnov,  I.V. Dunin-Barkovskii,  "Mathematische Statistik in der Technik" , Deutsch. Verlag Wissenschaft.  (1969)  (Translated from Russian)</TD></TR></table>

Revision as of 14:53, 7 June 2020

A non-parametric test for a hypothesis , according to which a random variable has a binomial distribution with parameters . If the hypothesis is true, then

where

and is the beta-function. According to the sign test with significance level , , the hypothesis is rejected if

where , the critical value of the test, is the integer solution of the inequalities

The sign test can be used to test a hypothesis according to which the unknown continuous distribution of independent identically-distributed random variables is symmetric about zero, i.e. for any real ,

In this case the sign test is based on the statistic

which is governed by a binomial law with parameters if the hypothesis is true.

Similarly, the sign test is used to test a hypothesis according to which the median of an unknown continuous distribution to which independent random variables are subject is ; to this end one simply replaces the given random variables by .

References

[1] 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)
[2] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[3] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
[4] N.V. Smirnov, I.V. Dunin-Barkovskii, "Mathematische Statistik in der Technik" , Deutsch. Verlag Wissenschaft. (1969) (Translated from Russian)
How to Cite This Entry:
Sign test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sign_test&oldid=48694
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article