GeneralisedNormalLaplace: Generalised Normal Laplace Distribution
In sjp/NormalLaplace: The Normal Laplace Distribution

Description Usage Arguments Details Value Author(s) References Examples

Density function, distribution function, quantiles and random number generation for the Generalised Normal Laplace distribution, with parameters mu (location), sigma (scale), alpha (shape), beta (shape) and rho (scale).

dgnl(x, mu = 0, sigma = 1, alpha = 1, beta = 1, rho = 1,
     param = c(mu, sigma, alpha, beta, rho))
pgnl(q, mu = 0, sigma = 1, alpha = 1, beta = 1, rho = 1,
     param = c(mu, sigma, alpha, beta, rho))
qgnl(p, mu = 0, sigma = 1, alpha = 1, beta = 1, rho = 1,
     param = c(mu, sigma, alpha, beta, rho),
     tol = 10^(-5), nInterpol = 100, subdivisions = 100, ...)
rgnl(n, mu = 0, sigma = 1, alpha = 1, beta = 1, rho = 1,
     param = c(mu, sigma, alpha, beta, rho))

`x, q`	Vector of quantiles.
`p`	Vector of probabilities.
`n`	Number of random variates to be generated.
`mu`	Location parameter mu, default is 0.
`sigma`	Scale parameter sigma, default is 1.
`alpha`	Shape parameter alpha, default is 1.
`beta`	Shape parameter beta, default is 1.
`rho`	Scale parameter rho, default is 1.
`param`	Specifying the parameters as a vector of the form `c(mu, sigma, alpha, beta, rho)`.
`tol`	Specified level of tolerance when checking if parameter beta is equal to 0.
`subdivisions`	The maximum number of subdivisions used to integrate the density and determine the accuracy of the distribution function calculation.
`nInterpol`	Number of points used in `qnl` for cubic spline interpolation of the distribution function.
`...`	Passes arguments to `uniroot`.

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 characteristic function of the distribution is required to be evaluted and integrated over in order to obtain the density and distribution functions.

The characteristic function is

phi(s) = ((alpha * beta * exp((- sigma^2 * s^2) / 2)) / ((alpha - is)*(beta + is)))^rho

The density function is

f(x - mu) = (1/pi) * int_0^inf r(s)*cos(theta(s) - s*y)ds

Where r(s) and θ(s) are respectively the modulus and argument of the complex number returned from the characteristic function.

The distribution function is

F(x - mu = (1/2) + ((1/pi) * int_0^inf (r(s)/s) * sin(s*x - theta(s))ds

Generation of random observations from the Generalised Normal Laplace distribution using rgnl is based upon the composition of three random variables, Z, G_1 and G_2. These are independent with Z ~ N(0, 1) and G_1, G_2 being gamma random variables with a scale of 1 and a shape of ρ.

A GNL random variable, X ~ GNL(mu, sigma, alpha, beta, rho) can be represented as:

X = rho*mu + sigma*sqrt(rho)*Z + (1/alpha)G_1 - (1/beta)G_2

dgnl gives the density function, pgnl gives the distribution function, qgnl gives the quantile function and rgnl generates random variates.

Simon Potter

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.

William J. Reed. (2008) Maximum Likelihood Estimation for Brownian-Laplace Motion and the Generalized Normal-Laplace (GNL) Distribution. Submitted to COMPSTAT 2008.

param <- c(0, 1, 3, 2, 1)
par(mfrow = c(1, 2))

## Curves of density and distribution
curve(dgnl(x, param = param), -5, 5, n = 1000)
title("Density of the Generalised Normal Laplace Distribution")
curve(pgnl(x, param = param), -5, 5, n = 1000)
title("Distribution Function of the Generalised Normal Laplace Distribution")

## Example of density and random numbers
par(mfrow = c(1, 1))
param1  <- c(0, 1, 1, 1, 1)
data1  <- rgnl(1000, param = param1)
curve(dgnl(x, param = param1),
      from = -5, to = 5, n = 1000, col = 2)
hist(data1, freq = FALSE, add = TRUE)
title("Density and Histogram")