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.

There are two versions to formulate an exponential dispersion model.

In the univariate case, a real-valued random variable ${\displaystyle X}${\displaystyle X} belongs to the additive exponential dispersion model with canonical parameter ${\displaystyle \theta }${\displaystyle \theta } and index parameter ${\displaystyle \lambda }${\displaystyle \lambda }, ${\displaystyle X\sim \mathrm {ED} ^{*}(\theta ,\lambda )}${\displaystyle X\sim \mathrm {ED} ^{*}(\theta ,\lambda )}, if its probability density function can be written as

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

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

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

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

In the multivariate case, the n-dimensional random variable ${\displaystyle \mathbf {X} }${\displaystyle \mathbf {X} } has a probability density function of the following form^[1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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