Cumulant

Let $\phi(t)$ be the Characteristic Function, defined as the Fourier Transform of the Probability Density Function,

$\begin{displaymath} \phi(t) = {\mathcal F}[P(x)] = \int_{-\infty}^\infty e^{itx}P(x)\,dx. \end{displaymath}$

(1)

Then the cumulants $\kappa_n$ are defined by

$\begin{displaymath} \ln\phi(t) \equiv \sum_{n=0}^\infty \kappa_n {(it)^n\over n!}. \end{displaymath}$

(2)

Taking the Maclaurin Series gives

$\ln\phi(t)=(it)\mu_1'+{\textstyle{1\over 2}}(it)^2(\mu_2'-{\mu'_1}^2)$
$\quad +{\textstyle{1\over 3!}}(it)^3(2{\mu'_1}^3-3\mu_1'\mu_2'+\mu_3')$
$\quad +{\textstyle{1\over 4!}}(it)^4(-6{\mu'_1}^4+12{\mu_1'}^2\mu_2'-3{\mu'_2}^2-4\mu_1'\mu_3'+\mu_4')$
$\quad +{\textstyle{1\over 5!}}(it)^5[-24{\mu'_1}^5+60{\mu'_1}^3\mu_2'+20{\mu'_1}^2\mu_3'+10\mu_2'\mu_3'$
$\quad +5\mu_1'(6{\mu'_2}^2-\mu_4')+\mu_5']+\ldots,$	(3)

where $\mu_n'$ are Moments about 0, so

$\displaystyle \kappa_1$	$\textstyle =$	$\displaystyle \mu_1'$	(4)
$\displaystyle \kappa_2$	$\textstyle =$	$\displaystyle \mu_2'-{\mu'_1}^2$	(5)
$\displaystyle \kappa_3$	$\textstyle =$	$\displaystyle 2{\mu'_1}^3-3\mu_1'\mu_2'+\mu_3'$	(6)
$\displaystyle \kappa_4$	$\textstyle =$	$\displaystyle -6{\mu'_1}^4+12{\mu_1'}^2\mu_2'-3{\mu'_2}^2-4\mu_1'\mu_3'+\mu_4'$	(7)
$\displaystyle \kappa_5$	$\textstyle =$	$\displaystyle -24{\mu'_1}^5+60{\mu'_1}^3\mu_2'+20{\mu'_1}^2\mu_3'+10\mu_2'\mu_3'+5\mu_1'(6{\mu'_2}^2-\mu_4')+\mu_5'.$	(8)

In terms of the Moments $\mu_n$ about the Mean,

$\displaystyle \kappa_1$	$\textstyle =$	$\displaystyle \mu$	(9)
$\displaystyle \kappa_2$	$\textstyle =$	$\displaystyle \mu_2=\sigma^2$	(10)
$\displaystyle \kappa_3$	$\textstyle =$	$\displaystyle \mu_3$	(11)
$\displaystyle \kappa_4$	$\textstyle =$	$\displaystyle \mu_4-3{\mu_2}^2$	(12)
$\displaystyle \kappa_5$	$\textstyle =$	$\displaystyle \mu_5-10\mu_2\mu_3,$	(13)

where $\mu$ is the Mean and $\sigma^2\equiv \mu_2$ is the Variance.

The k-Statistic are Unbiased Estimators of the cumulants.

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972.

Kenney, J. F. and Keeping, E. S. ``Cumulants and the Cumulant-Generating Function,'' ``Additive Property of Cumulants,'' and ``Sheppard's Correction.'' §4.10-4.12 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 77-82, 1951.