NormalLaplace (version 0.3-1)

NormalLaplaceMeanVar: 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))

Value

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

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).

Author

David Scott , Jason Shicong Fu

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

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)