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).
Remarks
- TDIST_XKURT is declared as deprecated. Please, use DIST_XKURT as TDIST_XKURT is listed here only for backward compatibility.
- 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:
- $ \nu\to 4^+$. $$\lim_{\nu\to 4^+}\gamma_2(\nu)=+\infty$$
- $ \nu\to \infty$. $$\lim_{\nu\to +\infty}\gamma_2(\nu)=0$$
Examples
Example 1:
Student's t-Distribution X-Kurtosis Plot:
Example 2:
| Formula | Description (Result) |
|---|---|
| =TDIST_XKURT(5) | Excess kurtosis with 5 degrees of freedom (6.000). |
| =TDIST_XKURT(100) | Student t-dist approaches Normality as v >> 1 (0.063). |
| =TDIST_XKURT(4.002) | Excess kurtosis increases as v approaches 4 (3000.000). |
Files Examples
Related Links
- Financial Dictionary - Excess kurtosis.
- Wikipedia - Excess kurtosis.
- Wikipedia - Student's t-distribution.
References
- 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.
- Hurst, Simon, The Characteristic Function of the Student-t Distribution, Financial Mathematics Research Report No. FMRR006-95, Statistics Research Report No. SRR044-95.
Comments
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