# Origins of Dispersion Models in Statistics [closed]

Recently, I have been reading about Dispersion Models in Statistics. For example, here is an example of the general form of a Dispersion Model:

In the univariate case, a real-valued random variable $$X$$ belongs to the additive exponential dispersion model with canonical parameter $$\theta$$ and index parameter $$\lambda$$, $$X \sim \text{ED}^*(\theta,\lambda)$$, if its probability density function can be written as

$$f_X(x \mid \theta,\lambda) = h^*(\lambda,x) \exp(\theta x - \lambda A(\theta)).$$

I have been trying to better understand the original reasons and the needs as to why Dispersion Models were developed. In particular, what kind problems did they attempt to solve? For example, in the following document (https://math.uoregon.edu/wp-content/uploads/2018/07/TempleStempleTweedieThesis.pdf) , it's stated that Dispersion Models are particularly useful at relating dissimilar probability distributions - for example, it is " nontrivial work to relate dissimilar distributions by just two parameters" (e.g. a Poisson Distribution and a Gamma Distribution).

My Question:

1) Does anyone know if the following statement is true - were Dispersion Models originally created to relate dissimilar distributions together? For example, if you were to sum a Poisson Distribution and a Gamma Distribution together, this sum might not satisfy the theoretical properties of a probability distribution. The attractive theoretical properties of the Normal Distribution ensures that affine (i.e. linear) sums of Normal Distributions still maintain a Normal Distribution - however, this result does not extend beyond Normal Distributions.

In a more official reference (https://rss.onlinelibrary.wiley.com/doi/pdfdirect/10.1111/j.2517-6161.1987.tb01685.x), a similar thought is expressed : "Based on techniques of combining and mixing exponential dispersion models, we present methods for constructing models for correlated data, such as longitudinal data or variance component problems. These models resemble, but do not include, the standard normal-theory models usually employed in these fields."

2) Is there some reference which demonstrates how different distributions from the family of Dispersion Models can be readily combined to create new distributions? Is there a generic function for combining distributions from the family of Dispersion Models?

I read the following reference (https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-017-0445-y) and came across a demonstration of this for the Tweedie Distribution (a probability distribution that belongs to the family of Dispersion Models):

But is there some general function that is used for combining probability functions from the family of Dispersion Models? Are there any general results that demonstrate that Dispersion Models can be readily combined to create new (and valid) probability distributions?

Thanks!

References: