new MultiLinkModel, dispersion modelling, new distribution families

[josef...@gmail.com]

I started to work on model classes for multiparameter distributions, especially models with  dispersion or heteroscedasticity that depends on explanatory variables. .

The last release included BetaModel for Beta regression with a link for the mean and a link for the precision (1 / variance) parameter.

The same can be extended to other models where variance or dispersion can depend on explanatory variables, Gaussian, T, Gamma, ...Negative Binomial, Beta-Binomial, ...

This uses full MLE but uses similar ideas as GLM does for one-parameter families.

Similar to GAMLSS, we can extend this to arbitrary distributions and an arbitrary number of distribution parameters and link functions. That's what the MultiLinkModel does.

My experimental and test case with many parameters is johnsonsu with 4 parameters.

The MultiLinkModel is similar to GenericLikelihoodModel and provides defaults so that only the loglikeobs of the distribution family needs to be specified. The rest relies on numerical derivatives and generic implementation.

It's targeted to being subclassed with additional distribution family specific information.

Currently, I have tried out Gaussian, johnsonsu, gamma and discrete distributions Beta-binomial and generalized poisson. 

Creating families with analytical derivatives, where possible, good starting parameters for optimization, and all the extras will be a lot of work.

This would be a good place to contribute if you have a favorite distribution where we don't have a regression model yet.

One of my higher priority families will be beta-binomial because we don't have an overdispersed alternative to binomial counts yet. 

We already have negative binomial and generalized poisson as overdispersed alternative to Poisson, and now it will be relatively easy to allow for dispersion that depends on explanatory variables.

We are missing parametric survival models like Weibull, and extreme value models, and ...

The next release, 0.14, sounds like fun.

Josef

[josef...@gmail.com]

I had forgotten to add the link to the PR, https://github.com/statsmodels/statsmodels/pull/7778 still dirty and WIP.

 

Josef