# From Marginal Exp-Norm Distributions to What Conditionals and Joint?

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?

• What do you mean by 'trivariate'? It would also help if you could to the extent possible use equations to illustrate your situation. Do you mean that you observe $Y$ where $Y = X + Z$, $X \sim Exp(\lambda)$ and $Z \sim N(\mu,\sigma^2)$? – user28 Oct 25 '10 at 16:47
• Sorry to use a non-standard term. The data are multivariate with three variables: R,G,B. I have the joint J(R,G,B) and I can compute the marginals. The form of the (univariate) marginals is the Y you defined. – MrDrFenner Oct 25 '10 at 18:31
• It would also help if you clarify a bit more what you mean by "understand this data". Is there a specific question(s) you are trying to answer or is this just exploratory data analysis? I would suggest editing the question to incorporate your comment and make the question as precise as possible so that useful answers can be given. – user28 Oct 25 '10 at 18:36
• I will try improve the question. I need to give it a bit of thought. -If- the data were multivariate normal (and, as follows, normal in the marginals), I would compute r and be done with it. So, maybe I have two options: (1) transform the data so they are normal or (2) compute r (via resampling) to assess its validity. As far as "understanding the data": ideally, I would want to characterize the conditional distributions (not necessarily by closed form equations) ... that is, I'd like to know how the variables affect each other. – MrDrFenner Oct 25 '10 at 19:29

## 1 Answer

Would copulas be any use here? I don't know enough about them, or your problem, to be sure.

• I know that copulas capture advanced correlation structures; I don't know much else about them. Unfortunately, I don't think they would be useful to me (for understanding the relationships in the data), nor to scientific colleagues (who would have to have some intuition/experience with copulas to understand my conclusions). – MrDrFenner Oct 25 '10 at 18:32
• @Mark In light of the revised question, I agree that looking at a copula-based description is a good suggestion. There's nothing terribly difficult about copulas: because you have the marginals, you can apply the probability integral transform to make the three variables (R, G, B) uniform on [0,1]. A copula merely describes their multivariate distribution in these terms. In effect, it eliminates the parameters from the marginals to allow you to focus on describing their multivariate dependencies. It affords the non-parametric approach you're asking for. – whuber Oct 26 '10 at 19:31
• Thanks for sticking with it through the reformulation. I will look into copulas. Will copulas allow me to compare marginals and conditionals from different Js: $J_1$, $J_2$, ... I might also wish to compare $R_1|G_1$ (from $J_1$) and $R_2|G_2$ in some way. The different joints are measurements on individuals in experimental conditions. Some of the measurements are repeated measurements of the same individual). – MrDrFenner Oct 26 '10 at 20:31
• @Mark Copulas will be useful for identifying, creating, and comparing models of joint distribution, but I don't think they will help in formal tests of hypothesis concerning those models. Consider asking another question in which you reveal more about the nature of the data and how they are collected, because that ought to play an important role in the modeling and choice of analytical techniques. – whuber Oct 26 '10 at 21:00
• Will do. It's always difficult to get the right abstraction of a problem. These two questions also seem to get at distribution free, multivariate analysis: stats.stackexchange.com/questions/4/… and stats.stackexchange.com/questions/1927/… – MrDrFenner Oct 26 '10 at 21:35