dist.Asymmetric.Log.Laplace {LaplacesDemon} | R Documentation |
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 |
scale |
This is the scale parameter |
kappa |
This is the asymmetry or skewness parameter
|
log |
Logical. If |
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.
Value
dallaplace
gives the density,
pallaplace
gives the distribution function,
qallaplace
gives the quantile function, and
rallaplace
generates random deviates.
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
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"))