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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.

vegetabletina
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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.

kjetil b halvorsen
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  • Thank you for your explanation! So can I think that the binomial distribution does not contain the dispersion parameter？I am pretty confused because I think the core of exponential dispersion models is the dispersion parameter! – vegetabletina Nov 16 '21 at 02:06
  • Also, I wonder if the definition of exponential dispersion family in your link is equal to that in Wikipedia? – vegetabletina Nov 16 '21 at 02:53
  • @vegetabletina It's also $1$ for the Poisson distribution. And the exponential distribution. – Glen_b Nov 16 '21 at 07:47
  • @Glen_b Is a(Φ) or Φ equal to 1? – vegetabletina Nov 16 '21 at 08:10
  • @vegetabletina For the binomial case (without observation-weights), the variance function is $\phi V(\mu)$ with $\phi=1$ and $V(\mu) = \mu(1-\mu)$. Just to be sure we're on the same page, what is the definition of $a(\phi)$ you're looking at? – Glen_b Nov 16 '21 at 10:53
  • @Glen_b I saw that from kjetil's link: https://stats.stackexchange.com/questions/413825/definition-of-exponential-family-with-dispersion-parameter?rq=1 – vegetabletina Nov 16 '21 at 11:44