Editing Generalized linear model (section)

=== Probability distribution ===
An '''overdispersed exponential family''' of distributions is a generalization of an [[exponential family]] and the [[exponential dispersion model]] of distributions and includes those families of probability distributions, parameterized by <math>\boldsymbol\theta</math> and <math>\tau</math>, whose density functions ''f'' (or [[probability mass function]], for the case of a [[discrete distribution]]) can be expressed in the form
:<math> f_Y(\mathbf{y} \mid \boldsymbol\theta, \tau) = h(\mathbf{y},\tau) \exp \left(\frac{\mathbf{b}(\boldsymbol\theta)^{\rm T}\mathbf{T}(\mathbf{y}) - A(\boldsymbol\theta)} {d(\tau)} \right). \,\!</math>

The ''dispersion parameter'', <math>\tau</math>, typically is known and is usually related to the variance of the distribution.  The functions <math>h(\mathbf{y},\tau)</math>, <math>\mathbf{b}(\boldsymbol\theta)</math>, <math>\mathbf{T}(\mathbf{y})</math>, <math>A(\boldsymbol\theta)</math>, and <math>d(\tau)</math> are known.  Many common distributions are in this family, including the normal, exponential, gamma, Poisson, Bernoulli, and (for fixed number of trials) binomial, multinomial, and negative binomial.

For scalar <math>\mathbf{y}</math> and <math>\boldsymbol\theta</math> (denoted <math>y</math> and <math>\theta</math> in this case), this reduces to
: <math> f_Y(y \mid \theta, \tau) = h(y,\tau) \exp \left(\frac{b(\theta)T(y) - A(\theta)}{d(\tau)} \right). \,\!</math>

<math>\boldsymbol\theta</math> is related to the mean of the distribution.  If <math>\mathbf{b}(\boldsymbol\theta)</math> is the identity function, then the distribution is said to be in [[canonical form]] (or ''natural form''). Note that any distribution can be converted to canonical form by rewriting <math>\boldsymbol\theta</math> as <math>\boldsymbol\theta'</math> and then applying the transformation <math>\boldsymbol\theta = \mathbf{b}(\boldsymbol\theta')</math>.  It is always possible to convert <math>A(\boldsymbol\theta)</math> in terms of the new parametrization, even if <math>\mathbf{b}(\boldsymbol\theta')</math> is not a [[one-to-one function]]; see comments in the page on [[exponential families]]. 

If, in addition, <math>\mathbf{T}(\mathbf{y})</math> and <math>\mathbf{b}(\boldsymbol\theta)</math> are the identity, then <math>\boldsymbol\theta</math> is called the ''canonical parameter'' (or ''natural parameter'') and is related to the mean through
:<math> \boldsymbol\mu = \operatorname{E}(\mathbf{y}) = \nabla_{\boldsymbol{\theta}} A(\boldsymbol\theta). \,\!</math>

For scalar <math>\mathbf{y}</math> and <math>\boldsymbol\theta</math>, this reduces to
:<math> \mu = \operatorname{E}(y) = A'(\theta).</math>

Under this scenario, the variance of the distribution can be shown to be<ref>{{harvnb|McCullagh|Nelder|1989}}, Chapter&nbsp;2.</ref>
:<math>\operatorname{Var}(\mathbf{y}) = \nabla^2_{\boldsymbol{\theta}} A(\boldsymbol\theta)d(\tau). \,\!</math>

For scalar <math>\mathbf{y}</math> and <math>\boldsymbol\theta</math>, this reduces to
:<math>\operatorname{Var}(y) = A''(\theta) d(\tau). \,\!</math>