GenLaplace() R function from [nlmm]

The Univariate Symmetric Generalized Laplace Distribution

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


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.

Returns

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

References

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

Author(s)

Marco Geraci

GenLaplace function