# Exponential dispersion model

In probability and statistics, the class of exponential dispersion models (EDM), also called exponential dispersion family (EDF), is a set of probability distributions that represents a generalisation of the natural exponential family. Exponential dispersion models play an important role in statistical theory, in particular in generalized linear models because they have a special structure which enables deductions to be made about appropriate statistical inference.

## Definition
### Univariate case

There are two versions to formulate an exponential dispersion model.

#### Additive exponential 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 $\lambda$, $X\sim \mathrm {ED} ^{*}(\theta ,\lambda )$, if its probability density function can be written as
    $f_{X}(x\mid \theta ,\lambda )=h^{*}(\lambda ,x)\exp \left(\theta x-\lambda A(\theta )\right)\,\!.$

#### Reproductive exponential dispersion model

The distribution of the transformed random variable $Y={\frac {X}{\lambda }}$ is called reproductive exponential dispersion model, $Y\sim \mathrm {ED} (\mu ,\sigma ^{2})$, and is given by
    $f_{Y}(y\mid \mu ,\sigma ^{2})=h(\sigma ^{2},y)\exp \left({\frac {\theta y-A(\theta )}{\sigma ^{2}}}\right)\,\!,$
with $\sigma ^{2}={\frac {1}{\lambda }}$ and $\mu =A'(\theta )$, implying $\theta =(A')^{-1}(\mu )$. The terminology dispersion model stems from interpreting $\sigma ^{2}$ as dispersion parameter. For fixed parameter $\sigma ^{2}$, the $\mathrm {ED} (\mu ,\sigma ^{2})$ is a natural exponential family.

### Multivariate case

In the multivariate case, the n-dimensional random variable $\mathbf {X}$ has a probability density function of the following form
    $f_{\mathbf {X} }(\mathbf {x} |{\boldsymbol {\theta }},\lambda )=h(\lambda ,\mathbf {x} )\exp \left(\lambda ({\boldsymbol {\theta }}^{\top }\mathbf {x} -A({\boldsymbol {\theta }}))\right)\,\!,$
where the parameter ${\boldsymbol {\theta }}$ has the same dimension as $\mathbf {X}$.

## Properties
### Cumulant-generating function

The cumulant-generating function of $Y\sim \mathrm {ED} (\mu ,\sigma ^{2})$ is given by
    $K(t;\mu ,\sigma ^{2})=\log \operatorname {E} [e^{tY}]={\frac {A(\theta +\sigma ^{2}t)-A(\theta )}{\sigma ^{2}}}\,\!,$
with $\theta =(A')^{-1}(\mu )$

### Mean and variance

Mean and variance of $Y\sim \mathrm {ED} (\mu ,\sigma ^{2})$ are given by
    $\operatorname {E} [Y]=\mu =A'(\theta )\,,\quad \operatorname {Var} [Y]=\sigma ^{2}A''(\theta )=\sigma ^{2}V(\mu )\,\!,$
with unit variance function $V(\mu )=A''((A')^{-1}(\mu ))$.

### Reproductive

If $Y_{1},\ldots ,Y_{n}$ are i.i.d. with $Y_{i}\sim \mathrm {ED} \left(\mu ,{\frac {\sigma ^{2}}{w_{i}}}\right)$, i.e. same mean $\mu$ and different weights $w_{i}$, the weighted mean is again an $\mathrm {ED}$ with
    $\sum _{i=1}^{n}{\frac {w_{i}Y_{i}}{w_{\bullet }}}\sim \mathrm {ED} \left(\mu ,{\frac {\sigma ^{2}}{w_{\bullet }}}\right)\,\!,$
with $w_{\bullet }=\sum _{i=1}^{n}w_{i}$. Therefore $Y_{i}$ are called reproductive.

### Unit deviance

The probability density function of an $\mathrm {ED} (\mu ,\sigma ^{2})$ can also be expressed in terms of the unit deviance $d(y,\mu )$ as
    $f_{Y}(y\mid \mu ,\sigma ^{2})={\tilde {h}}(\sigma ^{2},y)\exp \left(-{\frac {d(y,\mu )}{2\sigma ^{2}}}\right)\,\!,$
where the unit deviance takes the special form $d(y,\mu )=yf(\mu )+g(\mu )+h(y)$ or in terms of the unit variance function as $d(y,\mu )=2\int _{\mu }^{y}\!{\frac {y-t}{V(t)}}\,dt$.

## Examples

Many very common probability distributions belong to the class of EDMs, among them are: normal distribution, binomial distribution, Poisson distribution, negative binomial distribution, gamma distribution, inverse Gaussian distribution, and Tweedie distribution.