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Is there an extension of the generalized linear model that is able to estimate (using maximum likelihood) more than one parameter of a distribution at once. For example, I know that for a gamma distribution with shape parameter $k$ and scale parameter $\theta$, a GLM can estimate the coefficient matrix $\beta$ for predicting $\theta$ from given covariates, responses, and $k$. Is there a way to estimate both parameters as a linear combination of predictor variables ?

In chapter 10 of Generalized Linear Models (McCullagh & Nelder, 1989), Chapter 10 deals with joint modelling of mean and dispersion from covariates. However, they do not go into detail about a general algorithm to do so.

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It is certainly possible to do this with a linear model, but it no longer has the specific form of a generalised linear model. A good paper to read would be this one:

Peter M. Williams, Using Neural Networks to Model Conditional Multivariate Densities, Neural Computation, May 15, 1996, Vol. 8, No. 4, Pages 843-854 (www)

Which explains how to model a multivariate Gaussian as a function of some explanatory variables, including estimation of the covariance function of the multiple outputs. This basic idea also works with other distributions, for instance Williams published a paper in the late 90s on modelling rainfall data, where the outputs of the model are the probability of rain, and the shape and scale parameters of a Gamma distribution modelling the amount of rain that falls, given that it does rain. I've found this technique to be very useful in my work. The papers are published in terms of neural networks, but simplifying them for linear models is very easy - just omit the hidden layer of neurons.

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