distribution of the mean induced by a Dirichlet distribution

[Daniel Stutzbach]

Suppose I have a Dirichlet distribution, D, defined by parameters a1,
a2, ..., ak, which describe the Bayesian probability that p1, p2, ...,
pk are the parameters of some multinomial distribution, M.

The categories of the multinomial distribution represent monetary
values. For example, p3 represents the probability of $3. For this
reason, it makes sense to talk about the expected value of the
multinomial distribution, in the following sense:

E[M] = sum(1*p1 + 2*p2 + ... k*pk)

I would like to compute the distribution of E[M] from the Dirichlet
distribution. Is there an analytical solution to this problem?

Any help would be greatly appreciated,
Daniel Stutzbach, Ph.D.

[Herman Rubin]

In article <1190352012....@50g2000hsm.googlegroups.com>,

There is no problem with getting the mean and variance,
but I doubt that there is an exact solution with more
than two different multipliers. In that case, it is a
linear function of a Beta random variable.

Another way to see that it might be difficult is to use
the Gamma representation of the Dirichlet. If the
parameters are a_i, then if Z_i is Gamma(a_1, 1), the
pi can be generated by Z_i/\sum Z_j.

If b = \sum a_j is large, and all a_i are small compared
to b, the distribution should be approximately normal.

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558