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Fit Analyses

The Exponential Family of Distributions

The distribution of a random variable Y belongs to the exponential family if its probability (density) function can be written in the form

f(y ; \theta, \phi) = \exp( \frac{y \theta-b(\theta)}{a(\phi)} + c(y , \phi) )
where {\theta} is the natural or canonical parameter, \phi is the dispersion parameter, and a, b and c are specific functions.

The mean and variance of Y are then given by (McCullagh and Nelder 1989)

E(y) = \mu = {b'}(\theta)
\rm{Var}(y) = a(\phi) {b"}(\theta)

The function {{b"}(\theta)} can be expressed as a function of \mu, b"(\theta) = V(\mu),and it is called the variance function. Different choices of the function {b(\theta)}generate different distributions in the exponential family. For a binomial distribution with m trials, the function {a(\phi) = \phi/m}.For other distributions in the exponential family, {a(\phi) = \phi}.

SAS/INSIGHT software includes normal, inverse Gaussian, gamma, Poisson, and binomial distributions for the response distribution. For these response distributions, the density functions f(y), the variance functions {V(\mu)}, and the dispersion parameters \phi with function {a(\phi)} are

Normal
f(y) = \frac{1}{\sqrt{2 \pi} \sigma} \exp( -\frac{1}2( \frac{y-\mu}{\sigma} )^2 )    for -\infty\lt y\lt\infty

V(\mu) = 1

a(\phi) = \phi = {\sigma}^2

     

Inverse Gaussian
f(y) = \frac{1}{\sqrt{2\pi y^3} \sigma } \exp( -\frac{1}{2 \mu^2y} ( \frac{y-\mu}{{\sigma}} )^2 )    for y > 0

V(\mu) = \mu^3

a(\phi) = \phi = {\sigma}^2

     

Gamma
{f(y) = \frac{1}{y {\Gamma}({\nu})} (\frac{{\nu}y}{\mu})^{{\nu}} \exp(-\frac{{\nu}y}{\mu}) } {for y \gt 0}

{V(\mu) = \mu^2}

{a(\phi) = \phi = {\nu}^{-1}}

     

Poisson
{f(y) = \frac{\mu^y e^{-\mu}}{y! } } {for y=0, 1, 2, ... }

{V(\mu) = \mu}

{a(\phi) = \phi = 1}

     

Binomial
{f(y) = {m \choose r} \mu^r (1-\mu)^{m-r} } {for y= r/m, r=0, 1, 2,..., m}

{V(\mu) = \mu (1-\mu)}

{a(\phi) = \phi / m = 1/m}

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