From what I understand, the exponential family is defined as

$$f(y;\theta,\phi) = \exp\left(\frac{y\theta - b(\theta)}{a(\phi)}+c(y,\phi)\right) $$

I've read (but not seen shown anywhere), that the t distribution is not a member of the exponential family. But I don't understand why.

For instance, suppose I set $\theta = 0$, $b(\theta)=0$, and set $$c(y,\phi) = \ln\left(\frac{\Gamma\left(\frac{\phi+1}{2}\right)}{\sqrt{\phi\pi}\Gamma(\frac{\phi}{2})} \left(1+\frac{x^2}{\phi} \right)^{-\frac{\phi+1}{2}}\right)$$,

wouldn't then the t distribution then appear because

you would have

$$\exp\left(0+\ln\left(\frac{\Gamma\left(\frac{\phi+1}{2}\right)}{\sqrt{\phi\pi}\Gamma(\frac{\phi}{2})} \left(1+\frac{x^2}{\phi} \right)^{-\frac{\phi+1}{2}}\right)\right)=\frac{\Gamma\left(\frac{\phi+1}{2}\right)}{\sqrt{\phi\pi}\Gamma(\frac{\phi}{2})} \left(1+\frac{x^2}{\phi} \right)^{-\frac{\phi+1}{2}}$$

which is the t-distribution.

Why doesn't this work?


Ideally, I'd like to know why also my logic above is wrong (which I'm certain it is). So if you can fit that into your answer, that would be great.

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    $\begingroup$ Because the random variable and the parameter cannot separate as a product. $\endgroup$ – Xi'an Jan 12 '20 at 20:19
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    $\begingroup$ Your characterization of an exponential family is meaningless because every density function $f$ can be written in such a form: simply set $\theta=0=b(\theta)$ and $c(y;\phi)=\log f(y;\phi).$ $\endgroup$ – whuber Jan 12 '20 at 20:38
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    $\begingroup$ Your first equation is the definition of an exponential dispersion model, not the exponential family, and note that $a()$ is required to be positive. $\endgroup$ – Gordon Smyth Jan 13 '20 at 4:34
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    $\begingroup$ If $\phi$ is constant then it becomes a linear exponential family, but if $\theta$ is constant (as in your calculation) then it isn't anything at all. $\endgroup$ – Gordon Smyth Jan 13 '20 at 4:40

No, the t distribution is not an exponential family. Exponential family distributions do have existing moment generating functions, and the t distribution do not. See also Why doesn't the exponential family include all distributions?


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