LaplacesDemon (version 16.1.0)

dist.Asymmetric.Log.Laplace: Asymmetric Log-Laplace Distribution

Description

These functions provide the density, distribution function, quantile function, and random generation for the univariate, asymmetric, log-Laplace distribution with location parameter \(\mu\), scale parameter \(\lambda\), and asymmetry or skewness parameter \(\kappa\).

Usage

dallaplace(x, location=0, scale=1, kappa=1, log=FALSE)
pallaplace(q, location=0, scale=1, kappa=1)
qallaplace(p, location=0, scale=1, kappa=1)
rallaplace(n, location=0, scale=1, kappa=1)

Arguments

x, q

These are each a vector of quantiles.

p

This is a vector of probabilities.

n

This is the number of observations, which must be a positive integer that has length 1.

location

This is the location parameter \(\mu\).

scale

This is the scale parameter \(\lambda\), which must be positive.

kappa

This is the asymmetry or skewness parameter \(\kappa\), which must be positive.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Value

dallaplace gives the density, pallaplace gives the distribution function, qallaplace gives the quantile function, and rallaplace generates random deviates.

Details

• Application: Continuous Univariate

• Density 1: \(p(\theta) = \exp(-\mu)\frac{(\sqrt(2)\kappa / \lambda)(\sqrt(2) / \lambda\kappa)}{(\sqrt(2)\kappa / \lambda)+(\sqrt(2) / (\lambda\kappa))} \exp(-(\frac{\sqrt(2)\kappa}{\lambda})+1), \quad \theta \ge \exp(\mu)\)

• Density 2: \(p(\theta) = \exp(-\mu) \frac{(\sqrt(2)\kappa / \lambda) (\sqrt(2) / (\lambda\kappa))}{(\sqrt(2)\kappa / \lambda) + (\sqrt(2) / (\lambda\kappa))} \exp(\frac{\sqrt(2)(\log(\theta)-\mu)}{\lambda\kappa} - (\log(\theta)-\mu)), \quad \theta < \exp(\mu)\)

• Inventor: Pierre-Simon Laplace

• Notation 1: \(\theta \sim \mathcal{ALL}(\mu, \lambda, \kappa)\)

• Notation 2: \(p(\theta) = \mathcal{ALL}(\theta | \mu, \lambda, \kappa)\)

• Parameter 1: location parameter \(\mu\)

• Parameter 2: scale parameter \(\lambda > 0\)

• Mean: \(E(\theta) = \)

• Variance: \(var(\theta) = \)

• Mode: \(mode(\theta) = \)

The univariate, asymmetric log-Laplace distribution is derived from the Laplace distribution. Multivariate and symmetric versions also exist.

These functions are similar to those in the VGAM package.

References

Kozubowski, T. J. and Podgorski, K. (2003). "Log-Laplace Distributions". International Mathematical Journal, 3, p. 467--495.

See Also

dalaplace, dexp, dlaplace, dlaplacep, dllaplace, dmvl, dnorm, dnormp, dnormv.

Examples

# NOT RUN {
library(LaplacesDemon)
x <- dallaplace(1,0,1,1)
x <- pallaplace(1,0,1,1)
x <- qallaplace(0.5,0,1,1)
x <- rallaplace(100,0,1,1)

#Plot Probability Functions
x <- seq(from=0.1, to=10, by=0.1)
plot(x, dallaplace(x,0,1,0.5), ylim=c(0,1), type="l", main="Probability Function",
     ylab="density", col="red")
lines(x, dallaplace(x,0,1,1), type="l", col="green")
lines(x, dallaplace(x,0,1,5), type="l", col="blue")
legend(5, 0.9, expression(paste(mu==0, ", ", lambda==1, ", ", kappa==0.5),
     paste(mu==0, ", ", lambda==1, ", ", kappa==1),
     paste(mu==0, ", ", lambda==1, ", ", kappa==5)),
     lty=c(1,1,1), col=c("red","green","blue"))
# }