ABSTRACT
Dirichlet distribution given by (1.3) by introducing and
yielding the probability density
Let a random vector X = ( X i , . . . , Xm) ~ OD(a, 0). It easily fol lows from (IV. 1) that marginals distribution of Xi are given by
(in parameterization (II.1)), i = 1, . . . , m. The moments E[Xf\a,0] follow by substituting a f for a in (II.4). Analogously, the mean and the variance of Xi follow by substituting a f for a in (II.2) and (II.3), respectively. As above, the parameter 0 of the OD(a,0) may be interpreted as the common scale parameter amongst X = (X i, . . . , Xm), whereas the vector a + = ( a f , . . . , a:+) may be inter preted as a location parameter of X. Similarly to the analysis in Section A it follows that that the degenerate distribution with a point mass concentrated at parameter a + (cf. (IV.2)) is the degen erate distribution of an OD(a, 0) distribution by letting 0 —* oo. Letting 0 | 0 we deduce that the OD(a, 0) distribution converges to an ordered m-variate Bernoulli distribution with marginal pa rameters o+ (cf. (IV.2)), i = 1, . . . , m. The dependence structure in the limiting ordered m-variate Bernoulli distribution is obtained by studying the limiting behavior of the pairwise correlation coefficients in a OD(a, 0) distribution as 0 J. 0. Utilizing the reparameterization in (IV. 1) it follows that
and with (IV.3) and (II.4) we have
where, as above, |A| indicates the number of elements in the index set A and A(/) indicates the Zth element in A, such that A^ > k = 2 , . . . , |A|. Furthermore,
where. and
is a transformed Beta distribution with support
B . Solving for the Ordered Dirichlet Prior Param eters
To solve for the common shape parameter 0 and location pa rameter a = (q i, . . . , am) of an m dimensional random vector X ~ OD(a, 0) distribution using quantile estimates, we are required to solve problem Vz below.
