GED (Generalized Error Distribution)

Parameters

2. \(c \) (shape) and either \(b \) (scale) or \({\sigma ^2}\) (variance)

Kernel

\(\exp \left( {{ - {{\left( {|x|/b} \right)}^{2/c}}} \mathbin{/} 2 } \right)\)

Support

\(\left( -\infty ,\infty \right) \)

Mean

0 (a mean shift is simple, but rarely needed)

Variance

\(\frac{{{2^c}{b^2}\Gamma (3c/2)}}{{\Gamma (c/2)}}\)

Other Moments

\(E{\left| x \right|^r} = \frac{{{2^{rc/2}}{b^r}\Gamma ((r + 1)c/2)}}{{\Gamma (c/2)}}\)
 

\(E\frac{{\left| x \right|}}{\sigma } = \frac{{\Gamma (c)}}{{\sqrt {\Gamma (3c/2)\Gamma (c/2)} }}\)

Kurtosis

\(\frac{{\Gamma (5c/2)\Gamma (c/2)}}{{{{\left[ {\Gamma (3c/2)} \right]}^2}}}\)

This is greater than 3 (thus fat-tailed) if \(c > 1\) and less than 3 if \(c < 1\)

Main Uses

In financial econometrics, as an alternative to the t as the distribution of errors to provide different tail behavior from the Normal.

Density Function

%LOGGEDDENSITY(x,c,variance)

CDF

%GEDCDF(x,c) with variance pegged to 1.0

Inverse CDF

%INVGED(p,c) with variance pegged to 1.0

Random Draws

%RANGED(c,variance) (external function)

Special Cases

\(c=1\) is Normal, \(c=2\) is Laplace (two-tailed exponential)