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

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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

var(Y) = sigma^2 + 1/sigma^2 + 1/beta^2.

The skewness function is

skewness = [2/alpha^3 - 2/beta^3]/[sigma^2 + 1/alpha^2 + 1/beta^2]^3/2.

The kurtosis function is

kurtosis = [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 d.scott@auckland.ac.nz, 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. Birkh<e4>user, Boston.

Examples

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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)