Why is there no parameterization for heteroscedastic normal distribution?

Question

Very often in regression models, the assumption of homoscedascity is violated. I am wondering how is it possible that no one as of yet (as far as I could find in the literature) developed a parameterized heteroscedastic normal distribution?

So far, to assess heteroscedascity, researchers calculate the residuals of the model and then plot their variance across the predicted values of the outcome.

However, this seems like a "post-hoc" approach and it would be nicer if the heteroscedascity was already incorporated as a parameter in the parameterization of the distribution itself.

I would like to ask for reasons why this doesn't currently exist - was it tried, but attempts were unsuccessful? Is it impossible to model the heteroscedascity like this? Or is there another reason why this doesn't exist?

Answer 1

In statistics, a sequence (or a vector) of random variables is homoscedastic (/ˌhoʊmoʊskəˈdæstɪk/) if all its random variables have the same finite variance; this is also known as homogeneity of variance. The complementary notion is called heteroscedasticity, also known as heterogeneity of variance.

https://en.wikipedia.org/wiki/Homoscedasticity_and_heteroscedasticity

Homoscedastic errors mean that we are talking about random variables that have the same distributions. Heteroscedasticity means that they have different distributions. You cannot have distribution parametrized in a way that it can be different from itself.

There are though regression models that allow for heteroscedastic errors. Check for example the Generalized Least Squares or R’s nlme library that allows for fitting mixed-effects models with custom error variance functions.