In reply to Arnab Biswas.

Does this really come up in practice? I've never seen a 2-D distribution described in terms of quantiles of the 2-D CDF.

I think that is a challenging problem. I suspect that it is theoretically possible, but I have never done it myself. The problem is that you need a way to specify the multivariate CDF, which does not have a closed-form solution for most multivariate distributions. For example, I do not know an equation for the 2-D Dirichlet CDF. You can, of course, integrate the PDF by using SAS/IML software. If the quantile is known at (x1,x2), then for each vector of shape parameters, you need to approximate the CDF at (x1,x2) by the double integral
CDF(x_1, x_2) = \int_0^{x_1} \int_0^{x_2} f(x_1, x_2; \alpha_0, \alpha_1, \alpha_2) dx_1 dx_2
where {\alpha_0, \alpha_1, \alpha_2} are the unknown parameter values that you want to solve for by some method, such as nonlinear least squares estimation.