nlMeanVar: Mean, Variance, Skewness and Kurtosis of the Normal Laplace...

In NormalLaplace: The Normal Laplace Distribution

Mean, Variance, Skewness and Kurtosis of the Normal Laplace Distribution.

Description

Functions to calculate the mean, variance, skewness and kurtosis of a specified normal Laplace distribution.

Usage

nlMean(mu = 0, sigma = 1, alpha = 1, beta = 1,
         param = c(mu, sigma, alpha, beta))
nlVar(mu = 0, sigma = 1, alpha = 1, beta = 1,
         param = c(mu, sigma, alpha, beta))
nlSkew(mu = 0, sigma = 1, alpha = 1, beta = 1,
         param = c(mu, sigma, alpha, beta))
nlKurt(mu = 0, sigma = 1, alpha = 1, beta = 1,
         param = c(mu, sigma, alpha, beta))

Arguments

mu	Location parameter \mu, default is 0.
sigma	Scale parameter \sigma, default is 1.
alpha	Skewness parameter \alpha, default is 1.
beta	Shape parameter \beta, default is 1.
param	Specifying the parameters as a vector of the form c(mu, sigma, alpha, beta).

Details

Users may either specify the values of the parameters individually or as a vector. If both forms are specified, then the values specified by the vector param will overwrite the other ones.

The mean function is

E(Y)=\mu+1/\alpha-1/\beta.

The variance function is

V(Y)=\sigma^2+1/\alpha^2+1/\beta^2.%

The skewness function is

\Upsilon = [2/\alpha^3-2/\beta^3]/[\sigma^2+1/\alpha^2+1/\beta^2]^{3/2}.%

The kurtosis function is

\Gamma = [6/\alpha^4 + 6/\beta^4]/[\sigma^2+1/\alpha^2+1/\beta^2]^2.

Value

nlMean gives the mean of the skew hyperbolic nlVar the variance, nlSkew the skewness, and nlKurt the kurtosis.

Author(s)

David Scott [e-mail], Jason Shicong Fu

References

William J. Reed. (2006) The Normal-Laplace Distribution and Its Relatives. In Advances in Distribution Theory, Order Statistics and Inference, pp. 61–74. Birkhäuser, Boston.

Examples

param <- c(10,1,5,9)
nlMean(param = param)
nlVar(param = param)
nlSkew(param = param)
nlKurt(param = param)


curve(dnl(x, param = param), -10, 10)

NormalLaplace documentation built on July 4, 2024, 3 p.m.