dist.Asymmetric.Laplace {LaplacesDemon}  R Documentation 
Asymmetric Laplace Distribution: Univariate
Description
These functions provide the density, distribution function, quantile
function, and random generation for the univariate, asymmetric Laplace
distribution with location parameter location
, scale parameter
scale
, and asymmetry or skewness parameter kappa
.
Usage
dalaplace(x, location=0, scale=1, kappa=1, log=FALSE)
palaplace(q, location=0, scale=1, kappa=1)
qalaplace(p, location=0, scale=1, kappa=1)
ralaplace(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:
p(\theta) = \frac{\kappa \sqrt{2}}{\lambda (1+\kappa^2)} \exp(\theta\mu \frac{\sqrt{2}}{\lambda} \kappa^{\theta\mu} \theta\mu)
Inventor: Kotz, Kozubowski, and Podgorski (2001)
Notation 1:
\theta \sim \mathcal{AL}(\mu, \lambda, \kappa)
Notation 2:
p(\theta) = \mathcal{AL}(\theta  \mu, \lambda, \kappa)
Parameter 1: location parameter
\mu
Parameter 2: scale parameter
\lambda > 0
Parameter 3: skewness parameter
\kappa > 0
Mean:
E(\theta) = \mu + \lambda \frac{1/\kappa  \kappa}{\sqrt{2}}
Variance:
var(\theta) = \lambda^2 \frac{1 + \kappa^4}{2 \kappa^2}
Mode:
mode(\theta) = \mu
The asymmetric Laplace of Kotz, Kozubowski, and Podgorski (2001), also
referred to as AL, is an extension of the univariate, symmetric Laplace
distribution to allow for skewness. It is parameterized according to
three parameters: location parameter \mu
, scale parameter
\lambda
, and asymmetry or skewness parameter
\kappa
. The special case of \kappa=1
is the
symmetric Laplace distribution. Values of \kappa
in the
intervals (0, 1)
and (1, \infty)
,
correspond to positive (right) and negative (left) skewness,
respectively. The AL distribution is leptokurtic, and its kurtosis
ranges from 3 to 6 as \kappa
ranges from 1 to infinity. The
skewness of the AL has been useful in engineering and finance. As an
example, the AL distribution has been used as a replacement for
Gaussiandistributed GARCH residuals. There is also an extension to the
asymmetric multivariate Laplace distribution.
The asymmetric Laplace distribution is demonstrated in Kozubowski and Podgorski (2001) to be wellsuited for financial modeling, specifically with currency exchange rates.
These functions are similar to those in the VGAM
package.
Value
dalaplace
gives the density,
palaplace
gives the distribution function,
qalaplace
gives the quantile function, and
ralaplace
generates random deviates.
References
Kotz, S., Kozubowski, T.J., and Podgorski, K. (2001). "The Laplace Distribution and Generalizations: a Revisit with Applications to Communications, Economics, Engineering, and Finance". Boston: Birkhauser.
Kozubowski, T.J. and Podgorski, K. (2001). "Asymmetric Laplace Laws and Modeling Financial Data". Mathematical and Computer Modelling, 34, p. 10031021.
See Also
dlaplace
and
dallaplace
Examples
library(LaplacesDemon)
x < dalaplace(1,0,1,1)
x < palaplace(1,0,1,1)
x < qalaplace(0.5,0,1,1)
x < ralaplace(100,0,1,1)
#Plot Probability Functions
x < seq(from=5, to=5, by=0.1)
plot(x, dalaplace(x,0,1,0.5), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dalaplace(x,0,1,1), type="l", col="green")
lines(x, dalaplace(x,0,1,5), type="l", col="blue")
legend(1, 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"))