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\frac{1}{B ( a , b ) }
 
\frac{1}{B ( a , b ) }
  
\int\limits _ { 0 } ^ { z }  t  ^ {a-} 1 ( 1 - t )  ^ {b-} 1 dt ,\ \  
+
\int\limits _ { 0 } ^ { z }  t  ^ {a-1} ( 1 - t )  ^ {b-1} dt ,\ \  
 
0 \leq  z \leq  1 ,
 
0 \leq  z \leq  1 ,
 
$$
 
$$
  
 
and  $  B ( a , b ) $
 
and  $  B ( a , b ) $
is the beta-function. According to the sign test with significance level  $  \alpha $,  
+
is the [[beta-function]]. According to the sign test with significance level  $  \alpha $,  
 
$  0 < \alpha \leq  0 . 5 $,  
 
$  0 < \alpha \leq  0 . 5 $,  
 
the hypothesis  $  H _ {0} $
 
the hypothesis  $  H _ {0} $
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\frac \alpha {2}
 
\frac \alpha {2}
 
  ,\ \  
 
  ,\ \  
\sum _ {i = 0 } ^ { {m + 1 } \left ( \begin{array}{c}
+
\sum _ {i = 0 } ^ { {m + 1} } \left ( \begin{array}{c}
 
n \\
 
n \\
 
  i  
 
  i  
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$$  
 
$$  
{\mathsf P} \{ X _ {i} \langle - x \}  =  {\mathsf P} \{ X _ {i} \rangle x \} .
+
{\mathsf P} \{ X _ {i} < - x \}  =  {\mathsf P} \{ X _ {i} > x \} .
 
$$
 
$$
  
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$$  
 
$$  
\mu  =  \sum _ { i= } 1 ^ { n }  \delta ( X _ {i} ) ,\ \  
+
\mu  =  \sum _ { i=1} ^ { n }  \delta ( X _ {i} ) ,\ \  
\delta ( x)  =  \left \{
+
\delta ( x)  =  \left \{  
 +
\begin{array}{ll}
 +
1  & \textrm{ if }  x > 0 ,  \\
 +
0  & \textrm{ if }  x < 0 ,  \\
 +
\end{array}
 +
 
 +
\right .$$
  
 
which is governed by a binomial law with parameters  $  ( n ;  p = 0 . 5 ) $
 
which is governed by a binomial law with parameters  $  ( n ;  p = 0 . 5 ) $
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according to which the median of an unknown continuous distribution to which independent random variables  $  X _ {1} \dots X _ {n} $
 
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} $;  
 
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} $.
+
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>

Latest revision as of 17:10, 18 June 2020


A non-parametric test for a hypothesis $ H _ {0} $, 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

$$ I _ {z} ( a , b ) = \frac{1}{B ( a , b ) } \int\limits _ { 0 } ^ { z } t ^ {a-1} ( 1 - t ) ^ {b-1} dt ,\ \ 0 \leq z \leq 1 , $$

and $ B ( a , b ) $ 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

$$ \min \{ \mu , n - \mu \} \leq m , $$

where $ m = m ( \alpha , n ) $, the critical value of the test, is the integer solution of the inequalities

$$ \sum _ {i = 0 } ^ { m } \left ( \begin{array}{c} n \\ i \end{array} \right ) \left ( \frac{1}{2} \right ) ^ {n} \leq \frac \alpha {2} ,\ \ \sum _ {i = 0 } ^ { {m + 1} } \left ( \begin{array}{c} n \\ i \end{array} \right ) \left ( \frac{1}{2} \right ) ^ {n} > \frac \alpha {2} . $$

The sign test can be used to test a hypothesis $ H _ {0} $ 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 $,

$$ {\mathsf P} \{ X _ {i} < - x \} = {\mathsf P} \{ X _ {i} > x \} . $$

In this case the sign test is based on the statistic

$$ \mu = \sum _ { i=1} ^ { n } \delta ( X _ {i} ) ,\ \ \delta ( x) = \left \{ \begin{array}{ll} 1 & \textrm{ if } x > 0 , \\ 0 & \textrm{ if } x < 0 , \\ \end{array} \right .$$

which is governed by a binomial law with parameters $ ( n ; p = 0 . 5 ) $ if the hypothesis $ H _ {0} $ is true.

Similarly, the sign test is used to test a hypothesis $ H _ {0} $ 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

[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