16. A random variable Z defined on (0, ) is said to have a D-distribution with parameters , > 0 and k {0,

16. A random variable Z defined on (0, ∞) is said to have a D-distribution with parameters δ, β > 0 and k ∈ {0, 1, 2,...} if its density function is defined (up to a constant of proportionality) by pZ(z) ∝ zδ−1e−βz(1 − e−z)

k . (3.33)

This density emerges in many Bayesian nonparametric problems (e.g., Damien, Laud, and Smith, 1995). We wish to generate observations from this density.

(a) If Z’s density is log-concave, we know we can generate the necessary samples using the method of Gilks and Wild (1992). Find a condition (or conditions) under which this density is guaranteed to be log-concave.

(b) When Z’s density is not log-concave, we can instead use an auxiliary variable approach (Walker, 1995), adding the new random variable U = (U1,...,Uk). The Ui are defined on (0, 1) and mutually independent given Z, such that the joint density function of Z and U is defined (up to a constant of proportionality) by pZ,U (z, u) ∝ zδ−1e−βz 

k i=1 I(e−z,1)(ui) .

Show that this joint density function pZ,U does indeed have marginal density function pZ given in equation (3.33) above.

(c) Find the full conditional distributions for Z and U, and describe a sampling algorithm to obtain the necessary Z samples. What special subroutines would be needed to implement your algorithm?