kStatistics (version 2.0)

mom2cum: Moments in terms of cumulants

Description

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

Usage

mom2cum(n = 0)

Arguments

n

integer or vector of integers

Value

string

the expression of moments in terms of cumulants

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 moments are expressed in terms of cumulants 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, g[i]=k[i] for each i from 1 to n and k[i] cumulants. Multivariate moments are expressed in terms of multivariate cumulants 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 cumulants.

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 moment m[5] in terms of the cumulants k[1],...,k[5].
mom2cum(5)

# Multivariate moment m[3,1] in terms of the multivariate cumulants k[i,j] for 
# i=0,1,2,3 and j=0,1.
mom2cum(c(3,1))
# }