Asymmetric Log-Laplace Distribution

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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 λ, and asymmetry or skewness parameter kappa.

Usage

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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 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 1: p(theta) = exp(-mu) * (sqrt(2)*kappa/lambda) * (sqrt(2)/(lambda*kappa)) / ((sqrt(2)*kappa/lambda)+(sqrt(2)/(lambda*kappa))) * exp(-((sqrt(2)*kappa/lambda)+1)), theta >= exp(mu)

  • Density 2: p(theta) = exp(-mu) * (sqrt(2)*kappa/lambda) * (sqrt(2)/(lambda*kappa)) / ((sqrt(2)*kappa/lambda)+(sqrt(2)/(lambda*kappa))) * exp(((sqrt(2)*(log(theta)-mu)) / (lambda*kappa)) - (log(theta)-mu)), theta < exp(mu)

  • Inventor: Pierre-Simon Laplace

  • Notation 1: theta ~ ALL(mu, lambda, kappa)

  • Notation 2: p(theta) = 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

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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"))

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