GED_XKURT (Pro.)

	Attachment	Size
	GED_XKURT.xlsx

Calculates the excess kurtosis of the generalized error distribution (GEDⁱ).

Syntax

GED_XKURT(V)

V
is the shape parameter (or degrees of freedom) of the distribution (V > 1).

The generalized error distribution is also known as the exponential power distribution.
The probability density function of the GED is defined as:

$pdf(x)= \frac{e^{-\left |x \right |^\nu}}{2\Gamma(1+\frac{1}{\nu})}$

Where:
- $\nu$ is the shape parameter (or degrees of freedom).
The excess-kurtosis for GED(v) is defined as:

$\gamma_2= \frac{\Gamma (\frac{1}{\nu})\Gamma(\frac{5}{\nu})}{\Gamma(\frac{3}{\nu})^2}-3$

Where:
- $\Gamma (.)$ is the gamma function.
- $\nu$ is the shape parameter.
IMPORTANTThe GED excess kurtosis is only defined for shape parameters (degrees of freedom) greater than one.
Special Cases:
1. $\nu=2$
  
  GED becomes a normal distribution.
2. $\nu \to \infty$
  
  GED approaches uniform distribution.
  
  $\lim_{\nu \to \infty} \gamma_2(\nu) = -1.2$
3. $\nu \to 1^+$
  
  GED exhibits the highest excess-kurtosis (3).
  
  $\lim_{\nu \to 1^+}\gamma_2(\nu)=3$

GED Excess kurtosis plot

Example 1:

	A	B
1	Formula	Description (Result)
2	=GED_XKURT(2)	GED(2) is Normal distribution (0.000)
3	=GED_XKURT(1.0001)	Maximum excess kurtosis of a GED is 3.0 (3.000)
4	=GED_XKURT(100)	GED approaches uniform distribution for v >> 1 (-1.199)

Balakrishnan, N., Exponential Distribution: Theory, Methods and Applications, CRC, P 18 1996.