Distribution of dispersion submodel In double generalized linear models where we assume $y$ follows an exponential dispersion model, where the mean can be modelled as
$$g(\mu_i)=x_i^T\beta,$$
and the dispersion $(\phi)$ can be modelled as
$$h(\phi_i)=z_i^T\lambda$$
I have read in multiple articles that the deviance, which is used as response in the dispersion submodel can be assumed to be gamma distributed, so we can fit $\phi$ using a gamma GLM, but I do not understand why the unit deviance is gamma distributed.
 A: The saddle-point approximation ensures that the unit deviance is approximately $\phi$ times a $\chi^2_1$ random variable, see Section 5.4.3 of my book with Peter Dunn (Dunn & Smyth, 2018). The scaled $\chi^2_1$ distribution is a special case of a gamma distribution.
In the simplest case of a normal linear model, the unit deviances are just the squared residuals
$$d_i = (y_i-\mu_i)^2$$
In this case, standard normal linear assumptions imply that
$$d_i \sim \phi \chi^2_1$$
where $\phi$ is the variance of $y_i$.
However, in Smyth (1989) I showed that the gamma approximation for the unit deviance is exact only for normal and inverse-Gaussian generalized linear models. I derived the exact distribution of the unit deviances for a gamma generalized linear model and showed that it is a new distribution that I have called "digamma".
References
Dunn PK, Smyth GK (2018). Generalized linear models with examples in R. Springer, New York, NY. DOI: 10.1007/978-1-4419-0118-7. ISBN: 978-1-4419-0118-7.
Smyth, G. K. (1989). Generalized linear models with varying dispersion. Journal of the Royal Statistical Society B 51(1), 47-60.
Smyth, G. K., and Verbyla, A. P. (1999). Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics 10, 695-709.
http://www.statsci.org/smyth/pubs/Ties98-Preprint.pdf
Smyth, G. K., and Verbyla, A. P. (1999). Double generalized linear models: approximate REML and diagnostics. In Statistical Modelling: Proceedings of the 14th International Workshop on Statistical Modelling, Graz, Austria, July 19-23, 1999, H. Friedl, A. Berghold, G. Kauermann (eds.), Technical University, Graz, Austria, pages 66-80.
http://www.statsci.org/smyth/pubs/iwsm99-Preprint.pdf
