TOPICS
Search

Cumulant


Let phi(t) be the characteristic function, defined as the Fourier transform of the probability density function P(x) using Fourier transform parameters a=b=1,

phi(t)=F_x[P(x)](t)
(1)
=int_(-infty)^inftye^(itx)P(x)dx.
(2)

The cumulants kappa_n are then defined by

 lnphi(t)=sum_(n=1)^inftykappa_n((it)^n)/(n!)
(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 mu_n^' are raw moments, so

kappa_1=mu_1^'
(5)
kappa_2=mu_2^'-mu_1^'^2
(6)
kappa_3=2mu_1^'^3-3mu_1^'mu_2^'+mu_3^'
(7)
kappa_4=-6mu_1^'^4+12mu_1^'^2mu_2^'-3mu_2^'^2-4mu_1^'mu_3^'+mu_4^'
(8)
kappa_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^'.
(9)

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

In terms of the central moments mu_n,

kappa_1=mu
(10)
kappa_2=mu_2
(11)
kappa_3=mu_3
(12)
kappa_4=mu_4-3mu_2^2
(13)
kappa_5=mu_5-10mu_2mu_3,
(14)

where mu is the mean and sigma^2=mu_2 is the variance. These transformations can be given by CumulantToCentral[n].

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

kappa_(1,1)=-mu_(0,1)^'mu_(1,0)^'+mu_(1,1)^'
(15)
kappa_(2,1)=2mu_(0,1)^'mu_(1,0)^'^2-2mu_(1,0)^'mu_(1,1)^'-mu_(0,1)^'mu_(2,0)^'+mu_(2,1)^',
(16)

and central moments, e.g.,

kappa_(1,1)=mu_(1,1)
(17)
kappa_(2,1)=mu_(2,1)
(18)
kappa_(3,1)=-3mu_(1,1)mu_(2,0)+mu_(3,1)
(19)
kappa_(4,1)=-6mu_(2,0)mu_(2,1)-4mu_(1,1)mu_(3,0)+mu_(4,1)
(20)
kappa_(5,1)=30mu_(1,1)mu_(2,0)^2-10mu_(2,1)mu_(3,0)-10mu_(2,0)mu_(3,1)-5mu_(1,1)mu_(4,0)+mu_(5,1)
(21)

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

The k-statistics are unbiased estimators of the cumulants.


See also

Characteristic Function, Cumulant-Generating Function, Fourier Transform, k-Statistic, Kurtosis, Mean, Moment, Sheppard's Correction, Skewness, Unbiased Estimator, Variance

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Cumulant

Cite this as:

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

Subject classifications