kStatistics-package: Unbiased Estimators for Cumulant Products and Faa Di Bruno's...

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

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

Description

kStatistics is a package producing estimates of (joint) cumulants and (joint) cumulant products of a given dataset, using (multivariate) k-statistics and (multivariate) polykays, which are symmetric unbiased estimators. The procedures rely on a symbolic method arising from the classical umbral calculus and described in the referred papers. In the package, a set of combinatorial tools are given useful in the construction of these estimations such as integer partitions, set partitions, multiset subdivisions or multi-index partitions, pairing and merging of multisets. In the package, there are also functions to recover univariate and multivariate cumulants from a sequence of univariate and multivariate moments (and vice-versa), using Faa di Bruno's formula. The function producing Faa di Bruno's formula returns coefficients of exponential power series compositions such as f[g(z)] with f and g both univariate, or f[g(z1,...,zm)] with f univariate and g multivariate, or f[g1(z1,...,zm),...,gn(z1,...,zm)] with f and g both multivariate. Let us recall that Faa di Bruno's formula might also be employed to recover iterated (partial) derivatives of all these compositions. Lastly, using Faa di Bruno's formula, some special families of polynomials are also generated, such as Bell polynomials, generalized complete Bell polynomials, partition polynomials and generalized partition polynomials. Applications of these polynomials are described in the referred papers.

Author(s)

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

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

Maintainer: Giuseppe Guarino <[e-mail]>

References

C.A. Charalambides (2002) Enumerative Combinatoris, Chapman & Haii/CRC.

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 (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, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis. 52(11), 4909-4922. (download from https://arxiv.org/abs/0806.0129)

E. Di Nardo, G. Guarino, D. Senato (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing, 19, 155-165. (download from https://arxiv.org/abs/0807.5008)

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)

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

A. Nijenhuis, H. Wilf. (1978) Combinatorial Algorithms for Computers and Calculators. Academic Press, Orlando FL, II edition.

R. P. Stanley (2012) Enumerative combinatorics. Vol.1. II edition. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge.

Examples

# Some of the most important functions:

# Data assignment
data1<-c(16.34, 10.76, 11.84, 13.55, 15.85, 18.20, 7.51, 10.22, 12.52, 14.68, 
16.08, 19.43,8.12, 11.20, 12.95, 14.77, 16.83, 19.80, 8.55, 11.58, 12.10, 
15.02, 16.83, 16.98, 19.92, 9.47, 11.68, 13.41, 15.35, 19.11)

# Data assignment
data2<-list(c(5.31,11.16),c(3.26,3.26),c(2.35,2.35),c(8.32,14.34),c(13.48,49.45),
c(6.25,15.05),c(7.01,7.01),c(8.52,8.52),c(0.45,0.45),c(12.08,12.08),c(19.39,10.42))

# Return an estimate of the third cumulant of the random sample data1 with the indication 
# of which function has been employed
# KS:[1] -1.44706
nPolyk(c(3), data1, TRUE)

# Return an estimate of the product of the mean and the variance of the random sample data1
# with the indication of which function has been employed 
# PS:[1] 177.4233
nPolyk( list(  c(2), c(1) ), data1, TRUE)

# Return an estimate of the joint cumulant c[2,1] of the random sample data2  with the 
# indication of which function has been employed
# KM:[1] -23.7379
nPolyk(c(2,1), data2, TRUE);

# Return an estimate of the product of joint cumulants c[2,1]*c[1,0] of the random sample data2
# with the indication of which function has been employed
# PM:[1] 48.43243
nPolyk( list(  c(2,1), c(1,0) ), data2, TRUE)

# Return all the subdivisions of a multiset with only one element of multiplicity 3
mkmSet(3)

# Return all the subdivisions of a multiset with two elements, 
# having multiplicity respectively 2 and 1
mkmSet(c(2,1)) 
# OR (same output)
mkmSet(c(2,1), FALSE)

# Return the same output of the previous example but in a compact expression.
mkmSet(c(2,1), TRUE)  

# Return the scompositions of the vector (1,0,1) in 2 vectors of 3 non-negative integers 
# such that their sum gives (1,0,1), that is
# ([1,0,1],[0,0,0]) - ([0,0,0],[1,0,1]) - ([1,0,0],[0,0,1]) - ([0,0,1],[1,0,0]).
# Note that the second value in each resulting vector is always zero.
mkT(c(1,0,1),2) 
# OR (same output)
mkT(c(1,0,1),2, FALSE) 

# Return the same output of the previous example but in a compact expression.
mkT(c(1,0,1),2, TRUE) 

# Return all the partitions of the integer 4, that is 
# [1,1,1,1],[1,1,2],[1,3],[2,2],[4]
intPart(4) 
# OR (same output) 
intPart(4, FALSE) 

# Return the same output of the previous example but in a compact expression.
intPart(4, TRUE)

# Faa di Bruno's formula (Univariate with Univariate Case)
# The coefficient of z^2 in f[g(z)], that is f[2]g[1]^2 + f[1]g[2], where 
# f[1] is the coefficient of x in f(x) with x=g(z) 
# f[2] is the coefficient of x^2 in f(x) with  x=g(z) 
# g[1] is the coefficient of z in g(z)   
# g[2] is the coefficient of z^2 in g(z)   
# 
MFB( c(2), 1 )

# Faa di Bruno's formula (Univariate with Multivariate Case)    
# The coefficient of z1 z2 in f[g(z1,z2)], that is f[1]g[1,1] + f[2]g[1,0]g[0,1] 
# where 
# f[1]   is the coefficient of x in f(x) with x=g(z1,z2)
# f[2]   is the coefficient of x^2 in f(x) with x=g(z1,z2) 
# g[1,0] is the coefficient of z1 in g(z1,z2) 
# g[0,1] is the coefficient of z2 in g(z1,z2)  
# g[1,1] is the coefficient of z1z2 in g(z1,z2)  
# 
MFB( c(1,1), 1 )

# Faa di Bruno's formula (Multivariate with Multivariate Case) 
# The coefficient of z in f[g1(z),g2(z)], that is  f[1,0]g1[1] + f[0,1]g2[1] where 
# f[1,0] is the coefficient of x1 in f(x1,x2) with x1=g1(z) and x2=g2(z)
# f[0,1] is the coefficient of x2 in f(x1,x2) with x1=g1(z) and x2=g2(z)
# g1[1]  is the coefficient of z of g1(z)  
# g2[1]  is the coefficient of z of g2(z)  
MFB( c(1), 2 )

# The numerical value of f[1]g[1,1] + f[2]g[1,0]g[0,1], that is the coefficient of z1z2 
# in f[g1(z1,z2),g2(z1,z2)] output of MFB(c(1,1),1) when 
# f[1] = 5 and f[2] = 10 
# g[0,1]=3,  g[1,0]=6,  g[1,1]=9
e_MFB(c(1,1),1, c(5,10), c(3,6,9))

# 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))

# 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))

# The partition polynomial F[5]
pPart(5)

#  The general partition polynomial G[a1, a2; y1, y2], that is a2(y1^2) + a1(y2)
gpPart(2)

# The complete ordinary Bell Polynomial for n=5, that is 
# (y1^5) + 20(y1^3)(y2) + 30(y1)(y2^2) + 60(y1^2)(y3) + 120(y2)(y3) + 120(y1)(y4) + 120(y5)
oBellPol(5,)
# OR (same output)
oBellPol(5,0)

# The partial ordinary Bell polynomial for n=5 and m=3, that is 
# 30(y1)(y2^2) + 60(y1^2)(y3)
oBellPol(5,3)

# The complete exponential Bell Polynomial for n=5, that is 
# (y1^5) + 10(y1^3)(y2) + 15(y1)(y2^2) + 10(y1^2)(y3) + 10(y2)(y3) + 5(y1)(y4) + (y5)
eBellPol(5) 
# OR (same output)
eBellPol(5,0)

# The partial exponential Bell Polynomial for n=5 and m=3, that is 
# 15(y1)(y2^2) + 10(y1^2)(y3)
eBellPol(5,3)

# The Stirling number of second kind S(5,3) = 25   
e_eBellPol(5,3) 
# OR (same output) 
e_eBellPol(5,3,c(1,1,1,1,1))

# The Bell number B5 = 52   
e_eBellPol(5) 
# OR (same output) 
e_eBellPol(5,0)
# OR (same output)
e_eBellPol(5,0,c(1,1,1,1,1) )

# 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 )

# The generalized complete Bell Polynomial for n=1, m=2 and g1=g 
# that is 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 )

# The generalized complete Bell Polynomial for n=2, m=2 and g1=g2=g 
# that is (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 )

# The polynomial 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2)g[2,0]g[0,1], 
# output of GCBellPol( c(2,1),1 ), when g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5, 
# that is 16(y^2) + 4(y^3) + 5(y)
#
e_GCBellPol(c(2,1),1,,c(1:5))
#
# OR (same output)
#
e_GCBellPol(c(2,1),1,,c(1,2,3,4,5))
#
# OR (same output)
#
e_GCBellPol( c(2,1),1,"g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" )

# The polynomial 2(y^2)g[1,1]g[1,0] + (y^3)g[1,0]^2g[0,1] + (y)g[2,1] + (y^2)g[2,0]g[0,1], 
# output of GCBellPol( c(2,1),1 ) when g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5 and 
# y=7, that is 2191
#
e_GCBellPol( c(2,1),1,c(7),c(1:5) )
#
# OR (same output)
#
e_GCBellPol( c(2,1),1,"y=7, g[0,1]=1, g[1,0]=2, g[1,1]=3, g[2,0]=4, g[2,1]=5" )

 

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