What is the dispersion parameter of binomial distribution? Binomial distribution is a member of exponential dispersion models, but I can not find the dispersion parameter of it.
Could anyone help me find it out?
IMO the Binomial distribution only has an ordinary parameter p which describes both the mean and variance of this distribution, and I think there can't be the second parameter, neither does the dispersion parameter!
I find an answer in page 213 of Generalized Linear Models With Examples in R. But it changes the form of binomial distribution.
 A: Using the definition of an exponential dispersion family at Definition of exponential family with dispersion parameter, we find that for the binomial family it is $a(\phi)=1$. Copying that definition here:
$$
   f(y|θ,ϕ)=exp\left(yθ−b(θ)a(ϕ)+c(y,ϕ)\right)
$$
For the binomial distribution with mean $np=\mu$, say, the variance is $\sigma^2=np(1-p)=\mu(1-\mu/n)$, so is a function of the mean (assuming $n$ a fixed constant). Still, sometimes it is estimated as if a free parameter (if the binomial is seen as a model just for the mean, not for the dispersion, called under/over-dispersion). That must be done in the framework of quasi-likelihood, see Mean-variance relationship in the quasi-likelihood, What is the difference between logistic regression and Fractional response regression?
Giving more details, showing how the binomial probability mass function can be written in the above form:
\begin{align}
f(y \mid p) &= \binom{n}{y} p^y (1-p)^{n-y} \\
 &= \exp\left\{ \log\binom{n}{y} + y\log(p/(1-p)) - n\log(1-p)  \right\} \quad\text{Now writing $\theta=\log(p/(1-p))$} \\
 &= \exp\left\{ y\theta - n\log(1-p) + \log\binom{n}{y}  \right\} \\
&= \exp\left\{ \frac{y\theta + n\log(1+e^\theta)}{1} + \log\binom{n}{y}  \right\}
\end{align} making it clear that we have to choose $a(\phi)=1$ to match that particular definition.  In this case we could clearly simplify the definition, by replacing $a(\phi)$ by $\phi$, which is what is done in forinstance the R implementation glm for logistic regression.
