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''series of order statistics''
 
''series of order statistics''
  
An arrangement of the values of a random sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v0962701.png" /> with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v0962702.png" /> in ascending sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v0962703.png" />. The series is used to construct the empirical distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v0962704.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v0962705.png" /> is the number of terms of the series which are smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v0962706.png" />. Important characteristics of series of order statistics are its extremal terms (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v0962707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v0962708.png" />) and the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v0962709.png" />. The densities of the distributions of the minimum and maximum terms of a series of order statistics in the case
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An arrangement of the values of a random sample $  ( x _ {1} \dots x _ {n} ) $
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with distribution function $  F( x) $
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in ascending sequence $  x _ {(} 1) \leq  \dots \leq  x _ {(} n) $.  
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The series is used to construct the empirical distribution function $  {F _ {n} } ( x) = {m _ {x} } /n $,  
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where $  m _ {x} $
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is the number of terms of the series which are smaller than $  x $.  
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Important characteristics of series of order statistics are its extremal terms ( $  x _ {(} 1) = \min _ {1 \leq  i \leq  n }  x _ {i} $,  
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$  x _ {(} n) = \max _ {1 \leq  i \leq  n }  x _ {i} $)  
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and the range $  R _ {n} = {x _ {(} n) } - {x _ {(} 1) } $.  
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The densities of the distributions of the minimum and maximum terms of a series of order statistics in 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/v/v096/v096270/v09627010.png" /></td> </tr></table>
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$$
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F ( x)  = \int\limits _ {- \infty } ^ { x }  p ( y)  dy
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$$
  
 
are defined by the expressions
 
are defined by the expressions
  
<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/v096270/v09627011.png" /></td> </tr></table>
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$$
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p _ {(} 1) ( x)  = n [ 1 - F ( x)] ^ {n - 1 } p ( x)
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$$
  
 
and
 
and
  
<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/v096270/v09627012.png" /></td> </tr></table>
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$$
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p _ {(} n) ( x)  = nF ^ { n - 1 } ( x) p( x).
 +
$$
  
Considered as a stochastic process with time index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v09627013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096270/v09627014.png" />, the series of order statistics forms a non-homogeneous [[Markov chain|Markov chain]].
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Considered as a stochastic process with time index $  i $,  
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$  i = 1 \dots n $,  
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the series of order statistics forms a non-homogeneous [[Markov chain|Markov chain]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Wilks,  "Mathematical statistics" , Wiley  (1962)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Wilks,  "Mathematical statistics" , Wiley  (1962)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:28, 6 June 2020


series of order statistics

An arrangement of the values of a random sample $ ( x _ {1} \dots x _ {n} ) $ with distribution function $ F( x) $ in ascending sequence $ x _ {(} 1) \leq \dots \leq x _ {(} n) $. The series is used to construct the empirical distribution function $ {F _ {n} } ( x) = {m _ {x} } /n $, where $ m _ {x} $ is the number of terms of the series which are smaller than $ x $. Important characteristics of series of order statistics are its extremal terms ( $ x _ {(} 1) = \min _ {1 \leq i \leq n } x _ {i} $, $ x _ {(} n) = \max _ {1 \leq i \leq n } x _ {i} $) and the range $ R _ {n} = {x _ {(} n) } - {x _ {(} 1) } $. The densities of the distributions of the minimum and maximum terms of a series of order statistics in the case

$$ F ( x) = \int\limits _ {- \infty } ^ { x } p ( y) dy $$

are defined by the expressions

$$ p _ {(} 1) ( x) = n [ 1 - F ( x)] ^ {n - 1 } p ( x) $$

and

$$ p _ {(} n) ( x) = nF ^ { n - 1 } ( x) p( x). $$

Considered as a stochastic process with time index $ i $, $ i = 1 \dots n $, the series of order statistics forms a non-homogeneous Markov chain.

References

[1] S.S. Wilks, "Mathematical statistics" , Wiley (1962)

Comments

The phrase "variational series" is almost never used in the West. Cf. also Order statistic.

References

[a1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
How to Cite This Entry:
Variational series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variational_series&oldid=49127
This article was adapted from an original article by A.I. Shalyt (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article