kStatistics (version 2.0)

nKM: Multivariate K-Statistics

Description

Given a sample data, compute an estimate of the joint (multivariate) cumulant of a given order for the population distribution.

Usage

nKM( v, V)

Arguments

v

vector of integers

V

vector of sample data

Value

float

the value of the multivariate kstatistics of order v

Warning

The size of each value of the data vector must coincide with the length of the first parameter

Details

For a sample of i.i.d. random vectors, multivariate kstatistics are unbiased estimators of the population joint cumulants and are expressed in terms of the power sum symmetric polynomials in the random vectors of the sample. Thus, for the given multivariate sample data, nKM( c(r, s, ...), data) computes an estimate of the joint cumulant k[r, s, ...] of the population distribution.

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 http://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 http://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)

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

See Also

nKS, nPS, nPM

Examples

# NOT RUN {
# Data assignment
data1<-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 joint cumulant k[2,1]
nKM(c(2,1),data1)

# Data assignment
data2<-list(c(5.31,11.16,4.23),c(3.26,3.26,4.10),c(2.35,2.35,2.27),
c(4.31,10.16,6.45),c(3.1,2.3,3.2),c(3.20, 2.31, 7.3))

# Return an estimate of the joint cumulant k[2,2,2]
nKM(c(2,2,2),data2)
  
# Data assignment
data3<-list(c(5.31,11.16,4.23,4.22),c(3.26,3.26,4.10,4.9),c(2.35,2.35,2.27,2.26),
c(4.31,10.16,6.45,6.44),c(3.1,2.3,3.2,3.1),c(3.20, 2.31, 7.3,7.2))

# Return an estimate of the joint cumulant k[2,1,1,1]
nKM(c(2,1,1,1),data3)
 
# }