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Difference between revisions of "Studentized range"

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Let  $  X _ {1} \dots X _ {n} $
 
Let  $  X _ {1} \dots X _ {n} $
 
be independent normally  $  N( a, \sigma  ^ {2} ) $-
 
be independent normally  $  N( a, \sigma  ^ {2} ) $-
distributed random variables, and let  $  X  ^ {(} n) = ( X _ {(} n1) \dots X _ {(} nn) ) $
+
distributed random variables, and let  $  X  ^ {(n)} = ( X _ {(n1)} \dots X _ {(nn)} ) $
 
be the vector of order statistics constructed from the observations  $  X _ {1} \dots X _ {n} $.  
 
be the vector of order statistics constructed from the observations  $  X _ {1} \dots X _ {n} $.  
Moreover, let the statistic  $  \sum _ {i=} ^ {n} a _ {i} X _ {(} ni) $,  
+
Moreover, let the statistic  $  \sum _ {i=1} ^ {n} a _ {i} X _ {(ni)} $,  
which is a linear combination of the order statistics  $  X _ {(} n1) \dots X _ {(} nn) $,  
+
which is a linear combination of the order statistics  $  X _ {(n1)} \dots X _ {(nn)} $,  
 
be independent of some  "chi-squared"  distribution  $  V/ \sigma  ^ {2} $
 
be independent of some  "chi-squared"  distribution  $  V/ \sigma  ^ {2} $
 
of  $  f $
 
of  $  f $
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\frac{1}{s _ {f} }
 
\frac{1}{s _ {f} }
  \sum _ { i= } 1 ^ { n }  a _ {i} X _ {(} ni)
+
  \sum _ { i=1 } ^ { n }  a _ {i} X _ {(ni)}
 
$$
 
$$
  
 
is a Studentized statistic.
 
is a Studentized statistic.
  
The Studentized range is the Studentized statistic for which  $  \sum _ {i=} ^ {n} a _ {i} X _ {(} ni) $
+
The Studentized range is the Studentized statistic for which  $  \sum _ {i=1} ^ {n} a _ {i} X _ {(ni)} $
 
is the range of the sample  $  X _ {1} \dots X _ {n} $,  
 
is the range of the sample  $  X _ {1} \dots X _ {n} $,  
 
i.e. if
 
i.e. if
  
 
$$  
 
$$  
\sum _ { i= } 1 ^ { n }  a _ {i} X _ {(} ni)  =  X _ {(} nn) - X _ {(} n1) ;
+
\sum _ { i=1 } ^ { n }  a _ {i} X _ {(ni)} =  X _ {(nn)} - X _ {(n1)} ;
 
$$
 
$$
  
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$$  
 
$$  
  
\frac{X _ {(} nn) - X _ {(} n1) }{s _ {f} }
+
\frac{X _ {(nn)} - X _ {(n1)} }{s _ {f} }
 
  .
 
  .
 
$$
 
$$
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. David,  "Order statistics" , Wiley  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.S. Wilks,  "Mathematical statistics" , Wiley  (1962)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  H. David,  "Order statistics" , Wiley  (1970)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  S.S. Wilks,  "Mathematical statistics" , Wiley  (1962)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
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$$  
 
$$  
 
s _ {f}  ^ {2}  =   
 
s _ {f}  ^ {2}  =   
\frac{1}{n-}
+
\frac{1}{n-1} \sum ( X _ {i} - \overline{X}\; )  ^ {2}
1 \sum ( X _ {i} - \overline{X}\; )  ^ {2}
 
 
$$
 
$$
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Mood,  F.A. Graybill,  "Introduction to the theory of statistics" , McGraw-Hill  (1963)  pp. 243</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.H. Müller,  P. Neumann,  R. Storm,  "Tafeln der mathematischen Statistik" , C. Hauser  (1977)  pp. 166–169</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Mood,  F.A. Graybill,  "Introduction to the theory of statistics" , McGraw-Hill  (1963)  pp. 243</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.H. Müller,  P. Neumann,  R. Storm,  "Tafeln der mathematischen Statistik" , C. Hauser  (1977)  pp. 166–169</TD></TR>
 +
</table>

Latest revision as of 09:49, 20 July 2021


A statistic from the class of so-called Studentized statistics, obtained as a result of a special normalization of a linear combination of order statistics constructed from a normal sample.

Let $ X _ {1} \dots X _ {n} $ be independent normally $ N( a, \sigma ^ {2} ) $- distributed random variables, and let $ X ^ {(n)} = ( X _ {(n1)} \dots X _ {(nn)} ) $ be the vector of order statistics constructed from the observations $ X _ {1} \dots X _ {n} $. Moreover, let the statistic $ \sum _ {i=1} ^ {n} a _ {i} X _ {(ni)} $, which is a linear combination of the order statistics $ X _ {(n1)} \dots X _ {(nn)} $, be independent of some "chi-squared" distribution $ V/ \sigma ^ {2} $ of $ f $ degrees of freedom. Let $ s _ {f} ^ {2} = f ^ { - 1 } V $. In this case, one says that

$$ \frac{1}{s _ {f} } \sum _ { i=1 } ^ { n } a _ {i} X _ {(ni)} $$

is a Studentized statistic.

The Studentized range is the Studentized statistic for which $ \sum _ {i=1} ^ {n} a _ {i} X _ {(ni)} $ is the range of the sample $ X _ {1} \dots X _ {n} $, i.e. if

$$ \sum _ { i=1 } ^ { n } a _ {i} X _ {(ni)} = X _ {(nn)} - X _ {(n1)} ; $$

consequently, the Studentized range takes the form

$$ \frac{X _ {(nn)} - X _ {(n1)} }{s _ {f} } . $$

References

[1] H. David, "Order statistics" , Wiley (1970)
[2] S.S. Wilks, "Mathematical statistics" , Wiley (1962)

Comments

The case

$$ s _ {f} ^ {2} = \frac{1}{n-1} \sum ( X _ {i} - \overline{X}\; ) ^ {2} $$

is used for tests of normality and outlying observations, cf. [1], Chapt. 8. For a table of the quantiles of the Studentized range see [a2].

References

[a1] A.M. Mood, F.A. Graybill, "Introduction to the theory of statistics" , McGraw-Hill (1963) pp. 243
[a2] P.H. Müller, P. Neumann, R. Storm, "Tafeln der mathematischen Statistik" , C. Hauser (1977) pp. 166–169
How to Cite This Entry:
Studentized range. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Studentized_range&oldid=48884
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article