kStatistics (version 2.0)

cum2mom: Cumulants in terms of moments

Description

Compute simple and multivariate cumulants in terms of simple and multivariate moments.

Usage

cum2mom(n = 0)

Arguments

n

integer or vector of integers

Value

string

the expression of cumulants in terms of moments

Warning

The value of the first parameter is the same as the MFB function in the univariate with univariate case composition and in the univariate with multivariate case composition.

Details

Faa di Bruno's formula (the MFB function) gives the coefficients of the exponential formal power series obtained from the composition f(g()) of exponential formal power series. Simple cumulants are expressed in terms of moments using the Faa di Bruno's formula obtained from the MFB function in the case "composition of univariate f with univariate g" with f[i]=(-1)^(i-1)*(i-1)!, g[i]=m[i] for each i from 1 to n and m[i] moments. Multivariate cumulants are expressed in terms of multivariate moments using the Faa di Bruno's formula obtained from the MFB function in the case "composition of univariate f with multivariate g" whose coefficients are the multivariate moments.

References

E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. 14(2), 440-468. (download from http://arxiv.org/pdf/math/0607623.pdf)

E. Di Nardo E., G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286--6295. (download from http://arxiv.org/abs/1012.6008)

P. McCullagh, J. Kolassa (2009), Scholarpedia, 4(3):4699. http://www.scholarpedia.org/article/Cumulants

See Also

MFB

Examples

# NOT RUN {
# Simple cumulant k[5] in terms of the moments m[1],..., m[5].
cum2mom(5)

# Multivariate cumulant k[3,1] in terms of the multivariate moments m[i,j] for i=0,1,2,3 and j=0,1.
cum2mom(c(3,1))
# }