How can GAMLSS relax the GLM exponential family assumption?

Question

Generalized Additive Models for Location, Scale and Shape "relax the assumption of exponential families" in comparison to GLM's or GAM's. This is a direct quote from the paper by Rigby and Stasinopolous (2005). It does not say how they do this.

Now, I have read that GLM's also can be extended to allow for non-exponential family distributions. Is the original restriction just to make understanding the inference easier? Or is there some other approach by Rigby and S. which I am not seeing?

Answer 1

What Rigby and Stasinoplous' GAMLSS models allow is the modelling of all parameters of a distribution with separate linear predictors. Thomas Yee's Vector Generalised Additive Model (VGAM) class of models also allows for this kind of model to be fitted. More recently, there is a trend to calling such models Distributional Models because of the ability to model each parameter as a function of covariates.

Consider the negative binomial (NB) as a representative case. If $\theta$ the dispersion parameter is known and fixed then using the NB as the conditional distribution of the response falls into the realm of GLMs.

If we wish to estimate $\theta$ rather than set it to some known value, then that model does not fit into the GLM framework, but some software packages (I'm familiar with mgcv for R for example, plus glm.nb in the MASS package) allow estimation of $\theta$ whilst fitting.

In the case of estimated $\theta$ as mentioned above, $\theta$ is not estimated as a function of one or more covariates. In these models the covariates $X$ only affect the mean or expectation of the response.

In the GAMLSS, VGAM, or Distributional Models setting this restriction is relaxed. In the case of the NB, this means we might fit a model where the conditional distribution of the response is

$$y_i \sim \mathcal{NB}(\mu_i = \eta_{\mu, i}, \theta_i = \eta_{\theta, i})$$

with say

$$\eta_{\mu, i} = \alpha + \alpha_1 x_{1i} + \alpha_2 x_{2i}$$

(in which we model the $\mu$ parameter of the NB as an intercept plus linear effect of covariates $x_1$ and $x_2$) and

$$\eta_{\theta, i} = \beta + \beta_1 x_{2i} + \beta_3 x_{3i}$$

(in which we model the $\theta$ parameter of the NB as an intercept plus linear effect of covariates $x_2$ and $x_3$).

There is no reason that we must use the same set of covariates in each of the linear predictors.

In the context of estimating $\theta$, sensu mgcv or MASS packages, these would equate to intercept-only linear predictors for $\eta_{\theta, i}$

$$\eta_{\theta, i} = \beta$$

Another example is the Gaussian distribution. In the standard linear model the mean of the conditional distribution of the response is modelled by a linear combination of covariates, whilst the variance of the conditional distribution is the same for all observations and is just estimated from the model residuals. This is a special case of the GLM. A Gaussian location scale model allows both the mean and the variance of the conditional distribution of the response to be functions of covariates, eg:

$$y_i \sim \mathcal{N}(\mu_i = \eta_{\mu, i}, \sigma_i = \eta_{\sigma, i})$$

The general idea then with these kinds of approaches is to model, using covariates, all the parameters of the conditional distribution of the response.