kStatistics (version 1.0)

nKS: Simple K-Statistics

Description

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

Usage

nKS( v, V )

Arguments

v

integer or one-dimensional array

V

array of sample data

Value

float

the value of the kStatistics of order v

Details

For a sample of i.i.d. random variables, kstatistics are unbiased estimators of the population cumulants and are expressed in terms of the power sum symmetric polynomials in the random variables of the sample. Thus, for the given sample data, nKS(n,data) or nKS(c(n),data) computes an estimate of the n-th cumulant of the population distribution.

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

See Also

nPolyk, nKM, nPS, nPM

Examples

# NOT RUN {
data<-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)

nKS(7, data) 
# generate an estimate of the cumulant of order 7

nKS(1, data) 
# generate an estimate of the cumulant of order 1, that is the mean (R command: mean(data))

nKS(2, data) 
# generate an estimate of the cumulant of order 2, that is the variance (R command: var(data))

nKS(3, data)/sqrt(nKS(2, data))^3 
# generate an estimate of the skewness (R command: skewnes(data) in the library "moments")

nKS(4, data)/nKS(2, data)^2 + 3 
# generate an estimate of the kurtosis (R command: kurtosis(data) in the library "moments")
# }