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

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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 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.

Details

  • 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 asymmetric multivariate Laplace distribution.

The asymmetric Laplace distribution is demonstrated in Kozubowski and Podgorski (2001) to be well-suited 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. 1003-1021.

See Also

dlaplace and dallaplace

Examples

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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"))

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