Exponential dispersion model

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 ${\displaystyle X}$ belongs to the additive exponential dispersion model with canonical parameter ${\displaystyle \theta }$ and index parameter ${\displaystyle \lambda }$, ${\displaystyle X\sim {\mathrm{E}\mathrm{D}}^{\ast }(\theta ,\lambda )}$, if its probability density function can be written as

${\displaystyle f_X(x\mid \theta ,\lambda )=h^{\ast }(\lambda ,x)\exp \left(\theta x-\lambda A(\theta )\right).}$

Reproductive exponential dispersion model

The distribution of the transformed random variable ${\displaystyle Y=\frac{X}{\lambda }}$ is called reproductive exponential dispersion model, ${\displaystyle Y\sim \mathrm{E}\mathrm{D}(\mu ,{\sigma }^2)}$, and is given by

${\displaystyle f_Y(y\mid \mu ,{\sigma }^2)=h({\sigma }^2,y)\exp \left(\frac{\theta y-A(\theta )}{{\sigma }^2}\right),}$

with ${\displaystyle {\sigma }^2=\frac{1}{\lambda }}$ and ${\displaystyle \mu =A^{\prime }(\theta )}$, implying ${\displaystyle \theta =(A^{\prime }{)}^{-1}(\mu )}$. The terminology dispersion model stems from interpreting ${\displaystyle {\sigma }^2}$ as dispersion parameter. For fixed parameter ${\displaystyle {\sigma }^2}$, the ${\displaystyle \mathrm{E}\mathrm{D}(\mu ,{\sigma }^2)}$ is a natural exponential family.

Multivariate case

In the multivariate case, the n-dimensional random variable ${\displaystyle \mathbf{X}}$ has a probability density function of the following form

${\displaystyle f_{\mathbf{X}}(\mathbf{x}|\mathit{\theta },\lambda )=h(\lambda ,\mathbf{x})\exp \left(\lambda ({\mathit{\theta }}^{\mathrm{\top }}\mathbf{x}-A(\mathit{\theta }))\right),}$

where the parameter ${\displaystyle \mathit{\theta }}$ has the same dimension as ${\displaystyle \mathbf{X}}$.