Dirichlet distribution

A probability distribution on the simplex

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

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

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

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}, \ldots, 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

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

(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}, \ldots, \nu _ {k-1} = m _ {k-1} - m _ {k-2} $, $ \nu _ {k} = n - m _ {k-1} $.

References

[1]	S.S. Wilks, "Mathematical statistics" , Wiley (1962)
[a1]	T.S. Ferguson, "A Bayesian analysis of some nonparametric problems" Ann. Stat. , 1 (1973) pp. 209–230