# nlMeanVar: Mean, Variance, Skewness and Kurtosis of the Normal Laplace... In NormalLaplace: The Normal Laplace Distribution

## Description

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

## Usage

 ```1 2 3 4 5 6 7 8``` ```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 [email protected], 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

 ```1 2 3 4 5 6 7 8``` ```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 May 29, 2017, 12:23 p.m.