Exponential dispersion model

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) = 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. ==References==