# What is the "Generalized Error Distribution"?

In Nelson (1991), which develops the Exponential GARCH model, he refers (p. 352) to the "Generalized Error Distribution (GED)" and provides this density function:

$$f(z; \nu) = \frac{\nu\cdot\exp\left[ -\frac{1}{2}|z/\lambda|^{\nu} \right]}{\lambda 2^{(1+1/\nu)}\Gamma(1/\nu)},$$ where $$\begin{equation} \lambda\equiv \sqrt{\frac{2^{(-2/\nu)}\Gamma(1/\nu)}{\Gamma(3/\nu)}}, \end{equation}$$ and $$\Gamma(\cdot)$$ is the gamma function.

When $$\nu=2$$ this is equivalent to the standard Normal distribution, since $$\lambda=1$$ and

$$f(z; 2) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{z^2}{2}\right).$$

This distribution also appears in Leemis and McQueston (2008), which lists many distributions, where they call it the "Error (exponential power, general error)" distribution and write it as

$$f(z) = \frac{\exp\left[-\frac{1}{2}(|z-a|/b)^{2/c}\right]}{b(2^{c/2+1})\Gamma(1+c/2)}.$$

Letting $$a=0$$, $$b=\lambda$$, and $$c=2/\nu$$ we can see that these are the same distributions. Clearly $$a$$ is the mean parameter, which was assumed to be 0 in Nelson's paper.

I'm trying to understand the relation between this distribution and what Wikipedia calls the "generalized error distribution", with pdf

$$g(z) = \frac{\beta}{2\alpha\Gamma(1/\beta)} \; e^{-(|z-\mu|/\alpha)^\beta}.$$

Trying to write the Leemis and McQuestion version in the same form as Wikipedia, let $$\beta=2/c$$, $$a=\mu$$, and $$\alpha=b$$. We have

$$f(z) = \frac{\exp\left[-\frac{1}{2}(|z-\mu|/\alpha)^{\beta}\right]}{2\alpha(2^{1/\beta})\Gamma(1+1/\beta)} = \frac{\beta \exp\left[-\frac{1}{2}(|z-\mu|/\alpha)^{\beta}\right]}{2\alpha\Gamma(1/\beta)(2^{1/\beta})},$$

where I've used the fact that $$\Gamma(1+x)=x\Gamma(x)$$.

These do not look like the same distributions. There's an extra $$2^{1/\beta}$$ term in the denominator, and the $$-\frac{1}{2}$$ term in the exponential.

The question is, are these actually the same distributions? Have I made a mistake somewhere?

### References

Nelson, Daniel B. "Conditional heteroskedasticity in asset returns: A new approach." Econometrica (1991): 347-370.

Leemis, Lawrence M., and Jacquelyn T. McQueston. "Univariate distribution relationships." The American Statistician 62, no. 1 (2008): 45-53.

These are indeed the same distribution. I've found several parameterizations in use:

The Nelson (1991) parameterization is $$f(z) = \frac{\nu\cdot\exp\left[ -\frac{1}{2}|z/\lambda|^{\nu} \right]}{\lambda 2^{(1+1/\nu)}\Gamma(1/\nu)}.$$

To get the parameterization used on Wikipedia, let $$\nu=\beta$$ and $$\lambda=\alpha 2^{-1/\beta}$$, and add a mean parameter $$\mu$$, so

$$\begin{equation} f(z) = \frac{\beta}{2\alpha\Gamma(1/\beta)} \, \exp\left\{-\left(\frac{|z-\mu|}{\alpha}\right)^\beta\right\}. \end{equation}$$

where $$\mu$$, $$\alpha$$, and $$\beta$$ are the location, scale, and shape parameters, respectively.

Another parameterization, uses $$\beta=p$$ and $$\alpha=\sigma p^{1/p}$$. Then

\begin{align*} f(z) &= \frac{p}{2\sigma p^{1/p}\Gamma(1/p)} \, \exp\left\{-\frac{1}{p}\left(\frac{|z-\mu|}{\sigma}\right)^p\right\}, \\ &= \frac{1}{2\sigma p^{1/p}\Gamma(1+1/p)} \, \exp\left\{-\frac{1}{p}\left|\frac{z-\mu}{\sigma}\right|^p\right\}. \end{align*}

When the shape parameter, $$p=2$$, this reduces to a Normal distribution:

$$f(z) = \frac{1}{\sigma \sqrt{2\pi}} \, \exp\left\{-\frac{1}{2}\left(\frac{z-\mu}{\sigma}\right)^2\right\},$$

which follows from the fact that $$\Gamma(1/2)=\sqrt{\pi}$$. For a standard Normal with $$\sigma=1$$, we set $$\alpha=2^{1/2}=\sqrt{2}$$.