nPM {kStatistics}

Multivariate Polykays

Description

Given a multivariate data sample, the function returns an estimate of a product of joint cumulants with fixed orders.

Usage

nPM( v = NULL, V = NULL) 

Arguments

v

list of integer vectors

V

vector of a multivariate data sample

Details

Multivariate polykays or multivariate generalized k-statistics are unbiased estimators of joint cumulant products with minimum variance. See the referred papers to read more about these estimators. Multivariate polykays are usually expressed in terms of power sum symmetric polynomials in the random vectors of the sample. Thus, for the input multivariate sample data, running nPM( list( c(r1, s1, ...), c(r1, s2, ...),.. ), data) returns an estimate of the product k[r1, s1,....]*k[r2, s2, ...]*... where k[r1, s1,....], k[r2, s2, ...], ... are the joint cumulants of the population distribution.

Value

float

the estimate of the multivariate polykay

Warning

The size of each data vector must be equal to the length of the vector passed trough the first input variable. The vectors in the list must have the same length.

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

nPolyk, nKS, nKM, nPS

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