# How do you create a multivariate distribution with both continuous and discrete data?

I understand how to create a multivariate normal distribution to handle multiple sources of continuous data, and I understand how to create a multivariate categorical distribution to handle the discrete data. But I can't find any information on distributions over both continuous and discrete data at the same time.

Can someone point me to a book/paper/blog that describes how to do this?

• couldn't you just randomly sample from a multivariate normal and a multinomial and create a mixture distribution? Apologies, if you mean something else, I don't know a large amount about this area I'm afraid – richiemorrisroe Nov 13 '12 at 20:21
• You can use a copula to construct the joint distribution. – user10525 Nov 13 '12 at 20:23
• Take a look here: Ruscio, J., & Kaczetow, W. (2008). Simulating multivariate nonnormal data using an iterative technique. Multivariate Behavioral Research, 43, 355-381. – Valerian Nov 28 '12 at 11:01
• Can't you just break it into stages: first sample the categorical variables using a decision tree, then at each leaf of the tree sample your multivariate normal (or T, whatever). That's what we do in sampling demographic covariates for clinical trial simulation. – Mike Dunlavey Nov 28 '12 at 15:39

1. Simulate $(x_1,x_2,x_3)' \sim N(\mu,\Sigma)$, which you said you can do.
2. You can keep $y_1=x_1$ to get a continuous data.
3. You can generate $y_2 \sim {\rm Poisson}[ \exp(x_2) ]$, i.e., with the rate given by $\exp(x_2)$, to get count data.
4. You can generate $y_3 = \left\{ \begin{array}{ll} 1, & x_3<\kappa_1 \\ 2, & \kappa_1 < x_3 \le \kappa_2 \\ \vdots & \\ K-1, & x_3 > \kappa_K \end{array} \right.$ to get an ordinal variable (binary is a special case with $K=2$, although arguably a more traditional coding would be with 0 and 1 rather than 1 and 2).