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.^[1][2][3] 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 ${\displaystyle X}$ belongs to the additive exponential dispersion model with canonical parameter ${\displaystyle \theta }$ and index parameter ${\displaystyle \lambda }$, ${\displaystyle X\sim \mathrm {ED} ^{*}(\theta ,\lambda )}$, if its probability density function can be written as

${\displaystyle f_{X}(x\mid \theta ,\lambda )=h^{*}(\lambda ,x)\exp \left(\theta x-\lambda A(\theta )\right)\,\!.}$

Reproductive exponential dispersion model

The distribution of the transformed random variable ${\displaystyle Y={\frac {X}{\lambda }}}$ is called reproductive exponential dispersion model, ${\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma ^{2})}$, 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 ${\displaystyle \mu =A'(\theta )}$, implying ${\displaystyle \theta =(A')^{-1}(\mu )}$. The terminology dispersion model stems from interpreting ${\displaystyle \sigma ^{2}}$ as dispersion parameter. For fixed parameter ${\displaystyle \sigma ^{2}}$, the ${\displaystyle \mathrm {ED} (\mu ,\sigma ^{2})}$ 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^[1]

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

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

Properties

Cumulant-generating function

The cumulant-generating function of ${\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma ^{2})}$ is given by

${\displaystyle K(t;\mu ,\sigma ^{2})=\log \operatorname {E} [e^{tY}]={\frac {A(\theta +\sigma ^{2}t)-A(\theta )}{\sigma ^{2}}}\,\!,}$

with ${\displaystyle \theta =(A')^{-1}(\mu )}$

Mean and variance

Mean and variance of ${\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma ^{2})}$ are given by

${\displaystyle \operatorname {E} [Y]=\mu =A'(\theta )\,,\quad \operatorname {Var} [Y]=\sigma ^{2}A''(\theta )=\sigma ^{2}V(\mu )\,\!,}$

with unit variance function ${\displaystyle V(\mu )=A''((A')^{-1}(\mu ))}$.

Reproductive

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

${\displaystyle \sum _{i=1}^{n}{\frac {w_{i}Y_{i}}{w_{\bullet }}}\sim \mathrm {ED} \left(\mu ,{\frac {\sigma ^{2}}{w_{\bullet }}}\right)\,\!,}$

with ${\displaystyle w_{\bullet }=\sum _{i=1}^{n}w_{i}}$. Therefore ${\displaystyle Y_{i}}$ are called reproductive.

Unit deviance

The probability density function of an ${\displaystyle \mathrm {ED} (\mu ,\sigma ^{2})}$ can also be expressed in terms of the unit deviance ${\displaystyle d(y,\mu )}$ as

${\displaystyle f_{Y}(y\mid \mu ,\sigma ^{2})={\tilde {h}}(\sigma ^{2},y)\exp \left(-{\frac {d(y,\mu )}{2\sigma ^{2}}}\right)\,\!,}$

where the unit deviance takes the special form ${\displaystyle d(y,\mu )=yf(\mu )+g(\mu )+h(y)}$ or in terms of the unit variance function as ${\displaystyle d(y,\mu )=2\int _{\mu }^{y}\!{\frac {y-t}{V(t)}}\,dt}$.

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