dirichlet: The Dirichlet and generalized Dirichlet distribution In hyperdirichlet: A Generalization of the Dirichlet Distribution

Description

Specify a Dirichlet or generalized Dirichlet distribution as a special case of the hyperdirichlet distribution

Usage

 ```1 2 3 4 5``` ```dirichlet(params, powers, pnames) is.dirichlet(x) dirichlet_params(x) dirichlet_params(x) <- value gd(a, b, b0 = 0, pnames = NULL) ```

Arguments

 `params,powers` Numeric vectors (supply exactly one) specifying the parameters or the powers respectively of the Dirichlet distribution `x` Object of class hyperdirichlet `value` Numeric vector `a,b` Numeric vectors of the same length specifying the parameters of the generalized Dirichlet distribution `b0` Arbitrary constant for the generalized Dirichlet distribution `pnames` Character vector for name of the hyperdirichlet object

Details

Function `dirichlet()` returns the hyperdirichlet distribution corresponding to the classical Dirichlet distribution. If the vector `params|powers` is a named vector, then the hyperdirichlet object inherits the names (but the names are ignored if argument `pnames` is supplied).

Function `is.dirichlet(x)` returns `TRUE` or `FALSE` according to whether the hyperdirichlet object `x` is a Dirichlet distribution.

Function `dirichlet_params()` returns the Dirichlet parameters of a hyperdirichlet object.

Function `gd()` returns the hyperdirichlet distribution corresponding to the generalized Dirichlet distribution of Connor and Mosimann.

For convenience, the generalized Dirichlet distribution is described here. Connor and Mosimann 1969 give the PDF as

ommitted...see a LaTeXed file

where p_1+...+p_k=1 and b_0 is arbitrary. If b_{i-1}=a_i+b_i for i=2,...,k-1 then the PDF reduces to a standard Dirichlet distribution with alpha_i=a_i for i=1,...,k-1 and alpha_k=b_{k-1}.

Wong 1998 gives the algebraically equivalent form

ommitted...see a LaTeXed file

for x_1+...+x_k <= 1 and x_j >= 0 for j=1,2,...,k and gamma_j=beta_j-beta_{j+1} for j=1,2,...,k-1 and gamma_k=beta_{k-1}.

Here, B(x,y)=G(x)G(y)/G(x+y) is the beta function.

Value

Functions `dirichlet()` and `gd()` return a hyperdirichlet object; function `is.dirichlet()` returns a logical; and function `dirichlet_params()` returns a numeric vector.

Note

These functions have cheaply evaluated analytic expressions for the normalizing constant.

If a hyperdirichlet object corresponds to a Dirichlet distribution, this is relatively easy to detect [using `is.dirichlet()`]. However, the corresponding case for the generalized Dirichlet distribution is not yet coded up, owing to the non-neutrality of the GD.

Author(s)

Robin K. S. Hankin

References

• R. J. Connor and J. E. Mosimann 1969. Concepts of independence for proportions with a generalization of the Dirichlet distribution. Journal of the American Statistical Association, volume 64, number 325, pp194-206

• T-T Wong 1998. Generalized Dirichlet distribution in Bayesian analysis. Applied Mathematics and Computation, volume 97, pp165-181

`justpairs`
 ```1 2 3 4 5``` ```a <- dirichlet(1:4 , pnames=letters[1:4]) is.dirichlet(a) # should be TRUE dirichlet(dirichlet_params(a)) # should be 'a' gd(1:5,5:1) ```