Difference between revisions of "Sign test"

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

@@ Line 1: / Line 1: @@
-<!--
+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
-Automatically converted into TeX, above some diagnostics.
-Please remove this comment and the {{TEX|auto}} line below,
-if TeX found to be correct.
--->
-{{TEX|auto}}
+<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>
-{{TEX|done}}
-A [[Non-parametric test|non-parametric test]] for a hypothesis  $  H _ {0} $,
+<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
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085040/s0850407.png" /></td> </tr></table>
-I _ {z} ( a , b )  =
-\frac{1}{B ( a , b ) }
-\int\limits _ { 0 } ^ { z }  t  ^ {a-} 1 ( 1 - t )  ^ {b-} 1 dt ,\ \
-\leq  z \leq  1 ,
-$$
-and  $  B ( a , b ) $
+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} $
+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
-$$
+<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====
 <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>

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Difference between revisions of "Sign test"

Revision as of 14:53, 7 June 2020

References