I'm trying to write the chi-squared distribution in the form of: $$\exp\big\{(y\theta-b(\theta))/(\phi/w)-c(y,\phi)\big\}.$$

Using a shape-mean paramatization of the gamma distribution (i.e. taking shape parameter $\alpha$ and rate $\beta$, then re-parametrizing the $\beta$ parameter with $\mu=\alpha/\beta$, I was able to write the gamma distribution in this form, with: $$\theta=-1/\mu,\ b(\theta) = -\log(-\theta),\ \phi=1/\alpha,\ w=1,$$ and $$c(y,\phi) = \log(\Gamma(\alpha))-\alpha \log(\alpha)-(\alpha-1)\log(y).$$

As the chi-square is a special case of the gamma distribution with shape $\alpha=k/2$ and rate $\beta=1/2$, I initially tried using this. However I end up with $\theta=-1/k$ and $\phi=2/k$. In other words, the top and bottom of the first term in the exponential pdf form are literally just multiplied by $1/k$ in order to force a canonical parameter in there. This seems a bit weird to me - especially as if $\phi$ is a function of $\theta$, then $c(y,\phi)$ is a function of $\theta$ too, which it shouldn't be.

Other than that, I get a canonical parameter of $\theta=-1$. However, the $b()$ function is not meant to rely upon $\phi$, so there is no way of having $b'(\theta)=k$.

tldr: how do I go about expressing the chi-square pdf in the form of the exponential family of probability distribution?

Sorry if it's a bit hard to read! thanks!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.