# Parametric models for mixed discrete/continuous data

I'm curious if there are any common parametric distribution models for mixed discrete/continuous data. For illustration, suppose I have two random vectors, $$X_c,X_d$$, where $$X_c$$ is continuous and $$X_d$$ is discrete. I have data consisting of samples of $$(X_c,X_d)$$, and I'd like to do some density estimation. Ultimately the distribution of $$X_c\vert X_d$$ is what I would like, but I need to be able to vary $$X_d$$. I have a fair amount of data but it may be sparse in some areas, so it seems like a parametric model is a good place to start. But are there any standard parametric models for joint continuous/discrete data?

I could obviously go down the generative NN rabbit hole (GANs or VAEs w/ discrete-to-continuous encoding, for instance), but I'm curious about classical approaches as well.

• if you only need $P(X_c|X_d)$, then you don't actually need to do any joint modeling and can get away with just doing regression, and the classical approach is to use a (generalized) linear model. In these approaches, whether $X_d$ is continuous or discrete is not of primary importance. Commented May 5, 2023 at 19:48
• How many different values can $X_d$ take? Commented May 5, 2023 at 21:52
• I agree with John Madden. If you're interested in the conditional distribution, the distribution of $X_d$ doesn't matter, so your main issue is a choice (if you need one) of the conditional distribution(s) themselves, and tools related to regression and GLMs would be a good place to start Commented May 5, 2023 at 23:24

Parametric distributions for mixed discrete/continuous random variables can easily be built up as mixtures of standard parametric families of discrete distributions and continuous distributions. There are huge numbers of choices of mixtures you could make, depending on how many distributions you want to mix, and which ones you choose. In general, if you have any parametric discrete distribution with mass function $$p_\theta$$ and any parametric continuous distribution with density $$f_\phi$$ then you can create the corresponding parametric mixture distribution with CDF:
$$F(x|\theta, \phi, \lambda) = \lambda \sum_{r \leqslant x} p_\theta(r) + (1-\lambda) \int \limits_{0}^r f_\phi(r) \ dr.$$