dist.Log.Laplace {LaplacesDemon} R Documentation

## Log-Laplace Distribution: Univariate Symmetric

### Description

These functions provide the density, distribution function, quantile function, and random generation for the univariate, symmetric, log-Laplace distribution with location parameter location and scale parameter scale.

### Usage

dllaplace(x, location=0, scale=1, log=FALSE)
pllaplace(q, location=0, scale=1)
qllaplace(p, location=0, scale=1)
rllaplace(n, location=0, scale=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. log Logical. If log=TRUE, then the logarithm of the density is returned.

### Details

• Application: Continuous Univariate

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

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

• Inventor: Pierre-Simon Laplace

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

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

• Parameter 1: location parameter \mu

• Parameter 2: scale parameter \lambda > 0

• Mean: E(\theta) =

• Variance: var(\theta) =

• Mode: mode(\theta) =

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

These functions are similar to those in the VGAM package.

### Value

dllaplace gives the density, pllaplace gives the distribution function, qllaplace gives the quantile function, and rllaplace generates random deviates.

### References

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

dalaplace, dallaplace, dexp, dlaplace, dlaplacep, dmvl, dnorm, dnormp, and dnormv.

### Examples

library(LaplacesDemon)
x <- dllaplace(1,0,1)
x <- pllaplace(1,0,1)
x <- qllaplace(0.5,0,1)
x <- rllaplace(100,0,1)

#Plot Probability Functions
x <- seq(from=0.1, to=20, by=0.1)
plot(x, dllaplace(x,0,0.1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dllaplace(x,0,0.5), type="l", col="green")
lines(x, dllaplace(x,0,1.5), type="l", col="blue")
legend(2, 0.9, expression(paste(mu==0, ", ", lambda==0.1),
paste(mu==0, ", ", lambda==0.5), paste(mu==0, ", ", lambda==1.5)),
lty=c(1,1,1), col=c("red","green","blue"))


[Package LaplacesDemon version 16.1.6 Index]