nKM {kStatistics}

Multivariate K-Statistics

Description

Given a multivariate data sample, the function returns an estimate of a joint (or multivariate) cumulant with a fixed order.

Usage

nKM( v = NULL, V = NULL) 

Arguments

v

vector of integers

V

vector of a multivariate data sample

Details

For a sample of i.i.d. random vectors, multivariate k-statistics are unbiased estimators of the population joint cumulants with minimum variance and are expressed in terms of power sum symmetric polynomials in the random vectors of the sample. See the referred papers to read more about these estimators. Thus, for the input multivariate sample data, running nKM( c(r, s, ...), data) with fixed order v=(r, s, ...) returns an estimate of the joint cumulant k[r, s, ...] of the population distribution.

Value

float

the value of the multivariate k-statistics

Warning

The size of each data vector must be equal to the length of the vector passed trough the first input variable.

Note

Called by the master nPolyk 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, 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)

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

See Also

nKS, nPS, nPM

Examples

# 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)