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

Question

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?

Answer 1

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).