nKS {kStatistics}

Simple K-Statistics

Description

Given a data sample, the function returns an estimate of a cumulant with a fixed order.

Usage

nKS( v = NULL, V = NULL) 

Arguments

v

integer or one-dimensional vector

V

vector of a data sample

Details

For a sample of i.i.d. random variables, k-statistics are unbiased estimators with minimum variance of the population cumulants and are expressed in terms of power sum symmetric polynomials in the random variables of the sample. See the referred papers to read more about these estimators. Thus, for the input sample data, running nKS(v,data) or nKS(c(v),data) returns an estimate of the v-th cumulant of the population distribution.

Value

float

the value of the k-statistics

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, nKM, nPS, nPM

Examples

# 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