kStatistics (version 2.0)

nPM: Multivariate Polykays

Description

Given a multivariate sample data, the function returns an estimate of a product of population joint cumulants using multivariate polykays.

Usage

nPM( v, V )

Arguments

v

vector of integers

V

vector of multivariate sample data

Value

float

estimate of the multivariate polykay of order v

Warning

The size of each value of the data vector must coincide with the length of the first parameter. The vectors contained in the list must have the same length.

Details

Multivariate polykays are symmetric statistics which generalize polykays. They are unbiased estimators of products of joint cumulants of a distribution 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, nPM( list( c(r1, s1, ...), c(r1, s2, ...),.. ), data) computes an estimate of the product k[r1, s1,....]*k[r2, s2, ...]*... where k[r1, s1,....], k[r2, s2, ...], ... are joint cumulants 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

nPolyk, nKS, nKM, nPS

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 product k[2,1]*k[1,0]  where k[2,1] and k[1,0] are respectively 
# the cross-correlation of order (2,1) and the marginal mean of the population distribution 
nPM( list( c(2,1), c(1,0) ), 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 product k[2,0,1]*k[1,1,0] where k[2,0,1] and k[1,1,0] 
# are joint cumulants of the population distribution
nPM( list( c(2,0,1), c(1,1,0) ), data2) 

# }