dist.Asymmetric.Laplace: Asymmetric Laplace Distribution: Univariate
In benmarwick/LaplacesDemon: Complete Environment for Bayesian Inference

Description Usage Arguments Details Value References See Also Examples

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.

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)

`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.

Application: Continuous Univariate
Density: ((kappa*sqrt(2)) / (lambda*(1+kappa^2))) * exp(-|x-mu| * (sqrt(2)/lambda) * kappa^(|x-mu|) * |x-mu|)
Inventor: Kotz, Kozubowski, and Podgorski (2001)
Notation 1: theta ~ AL(mu, lambda, kappa)
Notation 2: p(theta) = 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*(1/kappa - kappa) / sqrt(2)
Variance: var(theta) = lambda^2 * (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, Inf), 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 Gaussian-distributed GARCH residuals. There is also an extension to the multivariate asymmetric Laplace distribution.

These functions are similar to those in the VGAM package.

dalaplace gives the density, palaplace gives the distribution function, qalaplace gives the quantile function, and ralaplace generates random deviates.

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.

dlaplace and dallaplace

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"))