GenLaplace {nlmm}

The Univariate Symmetric Generalized Laplace Distribution

Description

Density, distribution function, quantile function and random generation for the univariate symmetric generalized Laplace distribution.

Usage

dgl(x, sigma = 1, shape = 1, log = FALSE)
pgl(x, sigma = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)
qgl(p, sigma = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)
rgl(n, sigma = 1, shape = 1)

Arguments

x

vector of quantiles.

p

vector of probabilities.

n

number of observations.

sigma

positive scale parameter.

shape

shape parameter.

log, log.p

logical; if TRUE, probabilities are log–transformed.

lower.tail

logical; if TRUE (default), probabilities are P[X \le x] otherwise, P[X > x]. Similarly for quantiles.

Details

The univariate symmetric generalized Laplace distribution (Kotz et al, 2001, p.190) has density

f(x) = \frac{2}{\sqrt{2\pi}\Gamma(s)\sigma^{s+1/2}}(\frac{|x|}{\sqrt{2}})^{\omega}B_{\omega}(\frac{\sqrt{2}|x|}{\sigma})

where \sigma is the scale parameter, \omega = s - 1/2, and s is the shape parameter. \Gamma denotes the Gamma function and B_{u} the modified Bessel function of the third kind with index u. The variance is s\sigma^{2}.

This distribution is the univariate and symmetric case of MultivariateGenLaplace.

Value

dgl gives the density, pgl gives the distribution function, qgl gives the quantile function, and rgl generates random deviates.

Author(s)

Marco Geraci

References

Kotz, S., Kozubowski, T., and Podgorski, K. (2001). The Laplace distribution and generalizations. Boston, MA: Birkhauser.

See Also

MultivariateGenLaplace