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A [[Non-parametric test|non-parametric test]] for the homogeneity of two samples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v0960401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v0960402.png" />, based on the [[Rank statistic|rank statistic]]
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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/v/v096/v096040/v0960403.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v0960404.png" /> are the ranks (ordinal numbers) of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v0960405.png" /> in the series of joint order statistics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v0960406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v0960407.png" />; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v0960408.png" /> is defined by the pre-selected permutation
+
A [[Non-parametric test|non-parametric test]] for the homogeneity of two samples  $  Y _ {1} \dots Y _ {n} $
 +
and $  Z _ {1} \dots Z _ {m} $,
 +
based on the [[Rank statistic|rank 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/v/v096/v096040/v0960409.png" /></td> </tr></table>
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$$
 +
= \sum _ {i = 1 } ^ { m }
 +
\Psi \left (
 +
\frac{s ( r _ {i} ) }{m + n + 1 }
 +
\right ) ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604010.png" /> is the inverse function of the [[Normal distribution|normal distribution]] with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604011.png" />. The permutation is so chosen that for a given alternative hypothesis the test will be the strongest. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604012.png" />, irrespective of the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604014.png" /> individually, the asymptotic distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604015.png" /> is normal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604017.png" /> are independent and normally distributed with equal variances, the test for the alternative choice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604018.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604019.png" /> (in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096040/v09604020.png" />) is asymptotically equally as strong as the [[Student test|Student test]]. Introduced by B.L. van der Waerden [[#References|[1]]].
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where  $  r _ {i} $
 +
are the ranks (ordinal numbers) of the random variables  $  Z _ {i} $
 +
in the series of joint order statistics of  $  Y _ {j} $
 +
and  $  Z _ {i} $;
 +
the function  $  s( r) $
 +
is defined by the pre-selected permutation
 +
 
 +
$$
 +
\left ( \begin{array}{c}
 +
1 \\
 +
s ( 1)
 +
\end{array}
 +
\begin{array}{c}
 +
\dots \\
 +
\dots
 +
\end{array}
 +
 
 +
\begin{array}{c}
 +
( m + n) \\
 +
s ( m + n)
 +
\end{array}
 +
\right ) ,
 +
$$
 +
 
 +
and $  \Psi ( p) $
 +
is the inverse function of the [[Normal distribution|normal distribution]] with parameters $  ( 0, 1) $.  
 +
The permutation is so chosen that for a given alternative hypothesis the test will be the strongest. If $  m + n \rightarrow \infty $,  
 +
irrespective of the behaviour of $  m $
 +
and $  n $
 +
individually, the asymptotic distribution of $  X $
 +
is normal. If $  Y $
 +
and $  Z $
 +
are independent and normally distributed with equal variances, the test for the alternative choice $  {\mathsf P} ( Y < T) < {\mathsf P} ( Z < T) $
 +
or $  {\mathsf P} ( Y \langle  T) \rangle {\mathsf P} ( Z < T) $(
 +
in this case $  s( r) \equiv r $)  
 +
is asymptotically equally as strong as the [[Student test|Student test]]. Introduced by B.L. van der Waerden [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Order tests for the two-sample problem and their power"  ''Proc. Kon. Nederl. Akad. Wetensch. A'' , '''55'''  (1952)  pp. 453–458</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[3]</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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Order tests for the two-sample problem and their power"  ''Proc. Kon. Nederl. Akad. Wetensch. A'' , '''55'''  (1952)  pp. 453–458</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[3]</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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1979)</TD></TR></table>

Latest revision as of 08:27, 6 June 2020


A non-parametric test for the homogeneity of two samples $ Y _ {1} \dots Y _ {n} $ and $ Z _ {1} \dots Z _ {m} $, based on the rank statistic

$$ X = \sum _ {i = 1 } ^ { m } \Psi \left ( \frac{s ( r _ {i} ) }{m + n + 1 } \right ) , $$

where $ r _ {i} $ are the ranks (ordinal numbers) of the random variables $ Z _ {i} $ in the series of joint order statistics of $ Y _ {j} $ and $ Z _ {i} $; the function $ s( r) $ is defined by the pre-selected permutation

$$ \left ( \begin{array}{c} 1 \\ s ( 1) \end{array} \begin{array}{c} \dots \\ \dots \end{array} \begin{array}{c} ( m + n) \\ s ( m + n) \end{array} \right ) , $$

and $ \Psi ( p) $ is the inverse function of the normal distribution with parameters $ ( 0, 1) $. The permutation is so chosen that for a given alternative hypothesis the test will be the strongest. If $ m + n \rightarrow \infty $, irrespective of the behaviour of $ m $ and $ n $ individually, the asymptotic distribution of $ X $ is normal. If $ Y $ and $ Z $ are independent and normally distributed with equal variances, the test for the alternative choice $ {\mathsf P} ( Y < T) < {\mathsf P} ( Z < T) $ or $ {\mathsf P} ( Y \langle T) \rangle {\mathsf P} ( Z < T) $( in this case $ s( r) \equiv r $) is asymptotically equally as strong as the Student test. Introduced by B.L. van der Waerden [1].

References

[1] B.L. van der Waerden, "Order tests for the two-sample problem and their power" Proc. Kon. Nederl. Akad. Wetensch. A , 55 (1952) pp. 453–458
[2] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
[3] 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)

Comments

References

[a1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[a2] M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979)
How to Cite This Entry:
Van der Waerden test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Van_der_Waerden_test&oldid=49107
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article