I have (trivariate: multivariate with three variables) data that appears to be good empirical and reasonable theoretical fit for a (univariate) convolution of an exponential and a normal distribution (some times called exp-norm or exGauss distributions).
My data are samples from the joint distribution: J(R,G,B)
It appears that the marginals of R,G,B follow:
$R=R_N+R_E$ with $R_N \sim N(\mu_R,\sigma^2_R)$ and $R_E \sim Exp(\lambda_R)$
(and likewise for G,B).
I would like to effectively summarize the marginal, conditional, and joint distributions. The main purpose of the summary is to compare these distributions (marginal, conditional, and joint) with other distributions generated in a like manner. My difficulty is that (1) I don't know a form for the joint (or the conditional) distribution and (2) following from that, I have no parameters to estimate.
Thus, I need either a distribution free (non-parametric) approach to working with this data or I need to figure out a multivariate distribution that matches the data and the form of the marginals. Or should I think about other options?