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.

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 $$\mathbf{X}$$.

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 $$d(y,\mu) = y f(\mu) + g(\mu) + h(y)$$ 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.