mom2cum: Moments in terms of cumulants

In kStatistics: Unbiased Estimators for Cumulant Products and Faa Di Bruno's Formula

Moments in terms of cumulants

Description

The function compute a simple or a multivariate moment in terms of simple or multivariate cumulants.

Usage

mom2cum(n = 1)

Arguments

n	integer or vector of integers

Details

Faa di Bruno's formula (the MFB function) gives the coefficients of the exponential formal power series f[g()] where f and g are exponential formal power series too. Simple moments are expressed in terms of simple 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". In such a case the coefficients of g are the multivariate cumulants.

Value

string

the expression of the moment 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.

Note

This function calls the MFB function in the kStatistics package.

Author(s)

Elvira Di Nardo [e-mail],
Giuseppe Guarino [e-mail]

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 https://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 https://arxiv.org/abs/1012.6008)

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

See Also

MFB

Examples

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

# Return the 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))

kStatistics documentation built on June 8, 2022, 5:05 p.m.