What is the "Generalized Error Distribution"?

Question

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.

Answer 1

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 $\alpha=2^{1/2}=\sqrt{2}$.