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

## Syntax

**GED_XKURT**(

**V**)

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

## Remarks

- GED_XKURT is declared as deprecated. Please, use DIST_XKURT as GED_XKURT is listed here for backward compatibility.
- From time to time, it may be necessary for the definitions of some functions to be altered or removed from the add-in. In these circumstances the functions will first be declared deprecated and then removed from subsequent versions.
- 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.

**IMPORTANT**The GED excess kurtosis is only defined for shape parameters (degrees of freedom) greater than one.- Special Cases:

- $\nu=2$

GED becomes a normal distribution. - $\nu \to \infty$

GED approaches uniform distribution.

$$\lim_{\nu \to \infty} \gamma_2(\nu) = -1.2$$ - $\nu \to 1^+$

GED exhibits the highest excess-kurtosis (3).

$$\lim_{\nu \to 1^+}\gamma_2(\nu)=3$$

- $\nu=2$

## Examples

**Example 1: **

Formula | Description (Result) |
---|---|

=GED_XKURT(2) | GED(2) is Normal distribution (0.000) |

=GED_XKURT(1.0001) | Maximum excess kurtosis of a GED is 3.0 (3.000) |

=GED_XKURT(100) | GED approaches uniform distribution for v >> 1 (-1.199) |

**Example 2: **

### GED X-Kurtosis Plot

## Files Examples

## Related Links

## References

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

## Comments

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