# dist.Asymmetric.Log.Laplace: Asymmetric Log-Laplace Distribution In LaplacesDemon: Complete Environment for Bayesian Inference

## 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

 ```1 2 3 4``` ```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.

`dalaplace`, `dexp`, `dlaplace`, `dlaplacep`, `dllaplace`, `dmvl`, `dnorm`, `dnormp`, `dnormv`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```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")) ```