TDIST_XKURT (Pro.)

	Attachment	Size
	TDIST_XKURT.xlsx

Calculates the excess kurtosis of the Student's t-Distribution.

Syntax

TDIST_XKURT(v)

v
is the degrees of freedom of the Student's t-Distribution (v > 4).

The probability density function of the Student's t-Distribution is defined as:

$f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-(\nu+1)/2}$

Where:
- $\Gamma (.)$ is the gamma function.
- $\nu$ is the degrees of freedom (i.e. shape parameter).
The excess kurtosis of t-Distribution is defined as:

$\gamma_2= \frac{6}{\nu-4}$

Where:
- $\nu$ is the degrees of freedom.
IMPORTANT The Student's t-Distribution kurtosis is only defined for degrees of freedom values greater than 4.
Special Cases:
1. $\nu\to 4^+$
  
  $\lim_{\nu\to 4^+}\gamma_2(\nu)=+\infty$
2. $\nu\to \infty$
  
  $\lim_{\nu\to +\infty}\gamma_2(\nu)=0$

Student's t Excess kurtosis plot

Example 1:

	A	B
1	Formula	Description (Result)
2	=TDIST_XKURT(5)	Excess kurtosis with 5 degrees of freedom (6.000)
3	=TDIST_XKURT(100)	Student t-dist approaches Normality as v >> 1 (0.063)
4	=TDIST_XKURT(4.002)	Excess kurtosis increases as v approaches 4 (3000.000)

K.L. Lange, R.J.A. Little and J.M.G. Taylor. "Robust Statistical Modeling Using the t Distribution." Journal of the American Statistical Association 84, 881-896, 1989