< Probability distributions< *Exponential dispersion models*

## What are exponential dispersion models?

The **exponential dispersion family (EDF)** of probability distributions are special cases of exponential distributions that provide for a more convenient modeling framework. They are especially useful for fitting discrete data and show convenient properties, such as closure under convolution and infinite divisibility [1].

Many common distributions are EDFs, including the exponential distribution, gamma distribution and normal distribution. The EDF, along with the larger exponential family, are by the most important classes of distribution functions for regression modeling [2].

EDMs date back to the work of Maurice Tweedie in 1947 [3], where he outlined the structural properties of EDMs and outlined several special cases. EDMS were not popular until 25 years later, when Nelder and Wedderburn [4] proposed the EDF as the error distribution for their class of generalized linear models.

## Properties of exponential dispersion models

Exponential dispersion families have the probability density function (pdf)

Where

*θ*is the location parameter, also called the*natural parameter**ϕ*is the scale parameter, also called the*dispersion parameter.***I**t determines the spread of the distribution. In some cases, when the spread of the distribution is not of interest,*ϕ*is called a*nuisance parameter*[5].

## EDF members

Any member of the exponential family can be extended to EDFs. In the case of a single-parameter case, this is achieved via a transformation *Y* = *X*∕*ω*, where *ω* > 0 is a scaling factor and where *X* belongs to a single-parameter linear exponential family member, i.e., with *T*(*x*) = *x* [1]. The reason why this is true is because all members of the exponential family share a common moment generating function (MGF).

- Bernoulli distribution
- Beta distribution
- Binomial distribution
- Chi-squared distribution
- Exponential distribution
- Gamma distribution
- Inverse Gamma distribution
- Normal (Gaussian) distribution
- Poisson distribution.

## References

[1] Jørgensen, B. 1997. *The theory of exponential dispersion models, Monograph on statistics and probability*. Vol. 76. London: Chapman and Hall.

[2] Wüthrich, M.V., Merz, M. (2023). Exponential Dispersion Family. In: Statistical Foundations of Actuarial Learning and its Applications. Springer Actuarial. Springer, Cham. https://doi.org/10.1007/978-3-031-12409-9_2

[3] Tweedie, M. (1947). Functions of a statistical variate with given means, with special reference to Laplacian distributions. *Mathematical Proceedings of the Cambridge Philosophical Society,* *43*(1), 41-49. doi:10.1017/S0305004100023185

[4] Nelder, J. A., and R. W. M. Wedderburn. “Generalized Linear Models.” *Journal of the Royal Statistical Society. Series A (General)*, vol. 135, no. 3, 1972, pp. 370–84. *JSTOR*, https://doi.org/10.2307/2344614. Accessed 22 Aug. 2023.

[5] Chapter 6 Exponential Dispersion Family. Retrieved August 21, 2023 from: https://bookdown.org/ssjackson300/ASM_Lecture_Notes/exponential_dispersion_family.html