Cumulant -- from Wolfram MathWorld

Cumulant

Let be the characteristic function, defined as the Fourier transform of the probability density function using Fourier transform parameters ,

			(1)
			(2)

The cumulants are then defined by

(3)

(Abramowitz and Stegun 1972, p. 928). Taking the Maclaurin series gives

lnphi(t)=(it)mu_1^'+1/2(it)^2(mu_2^'-mu_1^'^2)+1/(3!)(it)^3(2mu_1^'^3-3mu_1^'mu_2^'+mu_3^')+1/(4!)(it)^4(-6mu_1^'^4+12mu_1^'^2mu_2^'-3mu_2^'^2-4mu_1^'mu_3^'+mu_4^')+1/(5!)(it)^5[24mu_1^'^5-60mu_1^'^3mu_2^'+20mu_1^'^2mu_3^'-10mu_2^'mu_3^'+5mu_1^'(6mu_2^'^2-mu_4^')+mu_5^']+...,

(4)

where are raw moments, so

			(5)
			(6)
			(7)
			(8)
			(9)

These transformations can be given by CumulantToRaw[n] in the Mathematica application package mathStatica.

In terms of the central moments ,

			(10)
			(11)
			(12)
			(13)
			(14)

where is the mean and is the variance. These transformations can be given by CumulantToCentral[n].

Multivariate cumulants can be expressed in terms of raw moments, e.g.,

			(15)
			(16)

and central moments, e.g.,

			(17)
			(18)
			(19)
			(20)
			(21)

using CumulantToRaw[m, n, ...] and CumulantToCentral[m, n, ...], respectively.

The k-statistics are unbiased estimators of the cumulants.

REFERENCES:

Abramowitz, M. and Stegun, I. 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.

CITE THIS AS:

Weisstein, Eric W. "Cumulant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cumulant.html