Namespaces
Variants
Actions

Difference between revisions of "Studentized range"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
s0907301.png
 +
$#A+1 = 15 n = 0
 +
$#C+1 = 15 : ~/encyclopedia/old_files/data/S090/S.0900730 Studentized range
 +
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}}
 +
{{TEX|done}}
 +
 
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s0907301.png" /> be independent normally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s0907302.png" />-distributed random variables, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s0907303.png" /> be the vector of order statistics constructed from the observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s0907304.png" />. Moreover, let the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s0907305.png" />, which is a linear combination of the order statistics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s0907306.png" />, be independent of some  "chi-squared"  distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s0907307.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s0907308.png" /> degrees of freedom. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s0907309.png" />. In this case, one says that
+
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
 +
 
 +
$$
  
<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/s090/s090730/s09073010.png" /></td> </tr></table>
+
\frac{1}{s _ {f} }
 +
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s09073011.png" /> is the range of the sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090730/s09073012.png" />, i.e. if
+
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
  
<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/s090/s090730/s09073013.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= } 1 ^ { n }  a _ {i} X _ {(} ni)  = X _ {(} nn) - X _ {(} n1) ;
 +
$$
  
 
consequently, the Studentized range takes the form
 
consequently, the Studentized range takes the form
  
<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/s090/s090730/s09073014.png" /></td> </tr></table>
+
$$
 +
 
 +
\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====
 
The case
 
The case
  
<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/s090/s090730/s09073015.png" /></td> </tr></table>
+
$$
 +
s _ {f}  ^ {2}  =
 +
\frac{1}{n-}
 +
1 \sum ( X _ {i} - \overline{X}\; )  ^ {2}
 +
$$
  
 
is used for tests of normality and outlying observations, cf. [[#References|[1]]], Chapt. 8. For a table of the quantiles of the Studentized range see [[#References|[a2]]].
 
is used for tests of normality and outlying observations, cf. [[#References|[1]]], Chapt. 8. For a table of the quantiles of the Studentized range see [[#References|[a2]]].

Revision as of 08:24, 6 June 2020


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=11560
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article