Multivariate Liouville Distributions
نویسنده
چکیده
A random vector (Xl' •.. , X n ), with positive components, is said to have a Liouville distribution if its joint probability density aC I an-l function is of the form f (xl + ... + x n ) xl •.. x n with the a i all positive. Examples of these are the Dirichlet and inverted Dirichlet distribution. In this paper, a comprehensive treatment of the Liouville distributions is provided. The results developed pertain to stochastic representations, transformation properties, complete neutrality, marginal and conditional distributions, regression functions, total positivity and reverse rule properties. Further, these topics are utilized in various characterizations of the Dirichlet and inverted Dirichlet distributions. Matrix analogs of the Liouville distributions are also defined, and many of the results obtained in the vector setting are extended appropriately.
منابع مشابه
Maximum entropy characterizations of the multivariate Liouville distributions
A random vector X = (X1, X2, . . . , Xn) with positive components has a Liouville distribution with parameter = ( 1, 2, . . . , n) if its joint probability density function is proportional to h( ∑n i=1 xi) ∏n i=1 x i−1 i , i > 0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233–256]. Examples include correlated gamma variables, Dirichlet and...
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