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A probability distribution on the simplex
 
A probability distribution on the simplex
  
<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/d/d032/d032840/d0328401.png" /></td> </tr></table>
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$$
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S _ {k}  = \{ {( x _ {1} \dots x _ {k} ) } : {x _ {1} \geq  0 \dots x _ {k} \geq  0 , x _ {1} + \dots + x _ {k} = 1 } \}
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,
 +
$$
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 +
where  $  k= 2, 3 \dots $
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determined by the probability density
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$$
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p ( x _ {1} \dots x _ {k} )  = \left \{
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\begin{array}{ll}
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C _ {k} \prod _ { i= } 1 ^ { k }  x _ {i} ^ {\nu _ {i} - 1 }  & \textrm{ if }  ( x _ {1} \dots x _ {k} ) \in S _ {k} ,  \\
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0 & \textrm{ if }  ( x _ {1} \dots x _ {k} ) \notin S _ {k} ,  \\
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\end{array}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d0328402.png" /> determined by the probability density
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\right .$$
  
<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/d/d032/d032840/d0328403.png" /></td> </tr></table>
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where  $  \nu _ {1} > 0 \dots \nu _ {k} > 0 $
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and
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d0328404.png" /> and
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$$
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C _ {k}  = \Gamma ( \nu _ {1} + \dots + \nu _ {k} ) \prod _ { i= } 1 ^ { k }
  
<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/d/d032/d032840/d0328405.png" /></td> </tr></table>
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\frac{1}{\Gamma ( \nu _ {i} ) }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d0328406.png" /> is the gamma-function. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d0328407.png" />, one has a special case of the Dirichlet distribution: the [[Beta-distribution|beta-distribution]]. The Dirichlet distribution plays an important role in the theory of order statistics. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d0328408.png" /> are independent random variables that are uniformly distributed over the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d0328409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d03284010.png" /> are the corresponding order statistics (cf. [[Order statistic|Order statistic]]), the joint distribution of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d03284011.png" /> differences
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where $  \Gamma ( \cdot ) $
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is the gamma-function. If $  k= 2 $,  
 +
one has a special case of the Dirichlet distribution: the [[Beta-distribution|beta-distribution]]. The Dirichlet distribution plays an important role in the theory of order statistics. For instance, if $  X _ {1} \dots X _ {n} $
 +
are independent random variables that are uniformly distributed over the interval $  [ 0, 1] $
 +
and $  X  ^ {(} 1) \leq  \dots \leq  X  ^ {(} n) $
 +
are the corresponding order statistics (cf. [[Order statistic|Order statistic]]), the joint distribution of the $  k $
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differences
  
<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/d/d032/d032840/d03284012.png" /></td> </tr></table>
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$$
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X ^ {( m _ {1} ) } , X ^ {( m _ {2} ) } - X ^ {( m _ {1} ) }
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\dots X ^ {( m _ {k-} 1 ) } - X ^ {( m _ {k-} 2 ) } , 1 - X ^
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{( m _ {k} ) }
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$$
  
(it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d03284013.png" />) has the Dirichlet distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d03284014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d03284015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032840/d03284016.png" />.
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(it is assumed that $  1 \leq  m _ {1} < m _ {2} < \dots < m _ {k-} 1 $)  
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has the Dirichlet distribution with $  \nu _ {1} = m _ {1} $,  
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$  \nu _ {2} = m _ {2} - m _ {1} \dots \nu _ {k-} 1 = m _ {k-} 1 - m _ {k-} 2 $,  
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$  \nu _ {k} = n - m _ {k-} 1 $.
  
 
====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====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.S. Ferguson,  "A Bayesian analysis of some nonparametric problems"  ''Ann. Stat.'' , '''1'''  (1973)  pp. 209–230</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.S. Ferguson,  "A Bayesian analysis of some nonparametric problems"  ''Ann. Stat.'' , '''1'''  (1973)  pp. 209–230</TD></TR></table>

Revision as of 19:35, 5 June 2020


A probability distribution on the simplex

$$ S _ {k} = \{ {( x _ {1} \dots x _ {k} ) } : {x _ {1} \geq 0 \dots x _ {k} \geq 0 , x _ {1} + \dots + x _ {k} = 1 } \} , $$

where $ k= 2, 3 \dots $ determined by the probability density

$$ p ( x _ {1} \dots x _ {k} ) = \left \{ \begin{array}{ll} C _ {k} \prod _ { i= } 1 ^ { k } x _ {i} ^ {\nu _ {i} - 1 } & \textrm{ if } ( x _ {1} \dots x _ {k} ) \in S _ {k} , \\ 0 & \textrm{ if } ( x _ {1} \dots x _ {k} ) \notin S _ {k} , \\ \end{array} \right .$$

where $ \nu _ {1} > 0 \dots \nu _ {k} > 0 $ and

$$ C _ {k} = \Gamma ( \nu _ {1} + \dots + \nu _ {k} ) \prod _ { i= } 1 ^ { k } \frac{1}{\Gamma ( \nu _ {i} ) } , $$

where $ \Gamma ( \cdot ) $ is the gamma-function. If $ k= 2 $, one has a special case of the Dirichlet distribution: the beta-distribution. The Dirichlet distribution plays an important role in the theory of order statistics. For instance, if $ X _ {1} \dots X _ {n} $ are independent random variables that are uniformly distributed over the interval $ [ 0, 1] $ and $ X ^ {(} 1) \leq \dots \leq X ^ {(} n) $ are the corresponding order statistics (cf. Order statistic), the joint distribution of the $ k $ differences

$$ X ^ {( m _ {1} ) } , X ^ {( m _ {2} ) } - X ^ {( m _ {1} ) } \dots X ^ {( m _ {k-} 1 ) } - X ^ {( m _ {k-} 2 ) } , 1 - X ^ {( m _ {k} ) } $$

(it is assumed that $ 1 \leq m _ {1} < m _ {2} < \dots < m _ {k-} 1 $) has the Dirichlet distribution with $ \nu _ {1} = m _ {1} $, $ \nu _ {2} = m _ {2} - m _ {1} \dots \nu _ {k-} 1 = m _ {k-} 1 - m _ {k-} 2 $, $ \nu _ {k} = n - m _ {k-} 1 $.

References

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

Comments

References

[a1] T.S. Ferguson, "A Bayesian analysis of some nonparametric problems" Ann. Stat. , 1 (1973) pp. 209–230
How to Cite This Entry:
Dirichlet distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_distribution&oldid=14736
This article was adapted from an original article by L.N. Bol'shev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article