kStatistics (version 2.0)

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 vector

V

vector 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. 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, nKM, nPS, nPM

Examples

# NOT RUN {

# Data assignment
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)

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

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

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

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

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

# }