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

(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.