kStatistics (version 1.0)

kStatistics-package: kStatistics

Description

kStatistics is a package allowing to estimate (joint)cumulants and products of (joint)cumulants of a given population distribution 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 devices are given such as integer partitions, set partitions and multiset subdivisions, pairing and merging of multisets, useful in the construction of these estimators.

References

E. Di Nardo, G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. Vol. 14(2), 440-468. (download from http://www.elviradinardo.it/lavori1.html)

E. Di Nardo, G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis Vol. 52(11), 4909-4922, (download from http://www.elviradinardo.it/lavori1.html)

E. Di Nardo, G. Guarino, D. Senato (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing, Vol. 19, 155-165. (download from http://www.elviradinardo.it/lavori1.html)

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

Examples

# NOT RUN {
# Some of the most important functions:

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)

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))

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

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

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

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

mkmSet(3)
# generate all subdivisions of a multiset with only one element of multiplicity 3
 
mkmSet(c(2,1)) 
# generate all subdivisions of a multiset with two elements, 
# having multiplicity respectively 2 and 1
 
# }