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

## 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

- Financial Dictionary - Excess kurtosis.
- Wikipedia - Excess kurtosis.
- Wikipedia - Exponential power distribution.
- GILLER, G.L, Generalized Error Distribution, working paper.

## References

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

## Comments

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