GCBellPol: Generalized Complete Bell Polynomial

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

Generalized Complete Bell Polynomial

Description

The function generates a generalized complete Bell polynomial, that is a coefficient of the composition exp(y[1] g1(z1,...,zm) + ... + y[n] gn(z1,...,zm)), where y[1],...,y[n] are variables. The input vector of integers identifies the subscript of the polynomial.

Usage

GCBellPol(nv = c(), m = 1, b = FALSE)

Arguments

nv	vector of integers, the subscript of the polynomial, corresponding to the powers of the product among z1, z2, ..., zm
m	integer, the number of z's variables
b	boolean, TRUE if the inner formal power series "g" are all equal

Details

The multivariate Faa di Bruno's formula, output of the MFB function, gives a coefficient of the multivariate exponential power series obtained from the composition of the multivariate exponential power series f(x1,...,xn) with xi=gi(z1,...,zm) for each i from 1 to n. Now, set f(y[1],...,y[n];x1,...,xn)=exp(y[1] x1 + ... + y[n] xn). In such a case, the coefficients are the generalized complete Bell polynomials, see the referred papers. In particular, the GCBellPol function gives the expression of these polynomials when n=1 or when n>1 and g1=...=gn=g or when n>1 and g1, ..., gn are all different. See the e_GCBellPol function for evaluating this polynomial when its variables y[1], ..., y[n] or/and its coefficients are substituted with numerical values.

Value

string

the expression of the polynomial

Warning

The value of the first parameter is the same as the mkmSet function.

Note

This function calls the MFB function in the kStatistics package.

Author(s)

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

References

G. M. Constantine, T. H. Savits (1996) A Multivariate Faa Di Bruno Formula With Applications. Trans. Amer. Math. Soc. 348(2), 503–520.

E. Di Nardo (2016) On multivariable cumulant polynomial sequence with applications. Jour. Algebraic Statistics 7(1), 72-89. (download from https://arxiv.org/abs/1606.01004)

E. Di Nardo, 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)

E. Di Nardo, M. Marena, P. Semeraro (2020) On non-linear dependence of multivariate subordinated Levy processes. In press Stat. Prob. Letters (download from https://arxiv.org/abs/2004.03933)

See Also

mkmSet, MFB, e_GCBellPol

Examples

# Return the generalized complete Bell Polynomial for n=1, m=1 and g1=g, 
# that is (y^2)g[1]^2 + (y)g[2]
#
GCBellPol( c(2),1 )

# Return the generalized complete Bell Polynomial for n=1, m=2 and g1=g, 
# 2(y^2)g[1,0]g[1,1] + (y^3)g[0,1]g[1,0]^2 + (y)g[2,1] + (y^2)g[0,1]g[2,0]
#
GCBellPol( c(2,1),1 )

# Return the generalized complete Bell Polynomial for n=2, m=2 and g1=g2=g, 
# (y1)g[1,1] + (y1^2)g[0,1]g[1,0] + (y2)g[1,1] + (y2^2)g[0,1]g[1,0] + 2(y1)(y2)g[0,1]g[1,0]
#
GCBellPol( c(1,1),2, TRUE )

# Return the generalized complete Bell Polynomial for n=2, m=2 and g1 different from g2, 
# that is (y1)g1[1,1] + (y1^2)g1[1,0]g1[0,1] + (y2)g2[1,1] + (y2^2)g2[1,0]g2[0,1] + 
# (y1)(y2)g1[1,0]g2[0,1] + (y1)(y2)g1[0,1]g2[1,0]
#
GCBellPol( c(1,1),2 )

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