cum2mom {kStatistics}

Cumulants in terms of moments

Description

The function computes a simple or a multivariate cumulant in terms of simple or multivariate moments.

Usage

cum2mom(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 cumulants are expressed in terms of simple 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 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". In such a case the coefficients of g are the multivariate moments.

Value

string

the expression of the cumulant 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.

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 cumulant k[5] in terms of the simple moments m[1],..., m[5].
cum2mom(5)

# Return the 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))