dirichlet function

Dirichlet distribution and generalizations

The Dirichlet distribution in likelihood (for p) form, including the generalized Dirichlet distribution due to Connor and Mosimann

dirichlet(powers, alpha)
GD(alpha, beta, beta0=0)
GD_wong(alpha, beta)
rdirichlet(n,H)
is.dirichlet(H)
rp_unif(n,H)

Arguments

• powers: In function dirichlet() a (named) vector of powers
• alpha,beta: A vector of parameters for the Dirichlet or generalized Dirichlet distribution
• beta0: In function GD(), an arbitrary parameter
• H: Object of class hyper2
• n: Number of observations

Details

These functions are really convenience functions.

Function rdirichlet() returns random samples drawn from a Dirichlet distribution. If second argument H is a hyper2 object, it is tested [with is.dirichlet()] for being a Dirichlet distribution. If so, samples from it are returned. If not, (e.g. icons), an error is given. If H is not a hyper2 object, it is interpreted as a vector of parameters $alpha$ [not a vector of powers].

Function rp_unif() returns uniformly distributed vectors, effectively using H*0; but note that this uses Dirichlet sampling which is much faster and better than the Metropolis-Hastings functionality documented at rp.Rd.

Functions GD() and GD_wong() return a likelihood function corresponding to the Generalized Dirichlet distribution as presented by Connor and Mosimann, and Wong, respectively. In GD_wong(), alpha and beta must be named vectors; the names of alpha give the names of $x_1,...,x_k$ and the last element of beta

gives the name of $x_{k+1}$.

References

• R. J. Connor and J. E. Mosimann 1969. Concepts of independence for proportions with a generalization of theDirichlet distribution . Journal of the American Statistical Association, 64:194--206
• T.-T. Wong 1998. Generalized Dirichlet distribution in Bayesian Analysis . Applied Mathematics and Computation, 97:165--181

Author(s)

Robin K. S. Hankin

Note

A dirichlet distribution can have a term with zero power. But this poses problems for hyper2 objects as zero power brackets are dropped.

See Also

hyper2,rp

Examples

x1 <- dirichlet(c(a=1,b=2,c=3))
x2 <- dirichlet(c(c=3,d=4))

x1+x2

H <- dirichlet(c(a=1,b=2,c=3,d=4))
rdirichlet(10,H)
colMeans(rdirichlet(1e4,H))