Exponential dispersion family

With dispersion $\varphi$ and known weight $w$, an exponential dispersion density has exponentIts mean is $b^{\prime }(\theta )$ and its variance is $(\varphi /w)b^{\prime \prime }(\theta )$.

The dispersion parameter scales the variance of an exponential dispersion family while leaving its mean function unchanged.

Underdispersion means less variation than prescribed by a reference statistical model, corresponding to a working dispersion parameter below one. It is not, by itself, evidence that the reference mean function is improved.

Overdispersion occurs when observed conditional variation exceeds that prescribed by a statistical model. For count data it commonly means variance greater than the mean assumed by a Poisson model.

The variance function writes an exponential dispersion family's variance as $\mathrm{Var}(Y)=\varphi V(\mu )/w$.