# dist.Skew.Laplace: Skew-Laplace Distribution: Univariate In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

These functions provide the density, distribution function, quantile function, and random generation for the univariate, skew-Laplace distribution with location parameter mu, and two mixture parameters: alpha and beta.

## Usage

 ```1 2 3 4``` ```dslaplace(x, mu, alpha, beta, log=FALSE) pslaplace(q, mu, alpha, beta) qslaplace(p, mu, alpha, beta) rslaplace(n, mu, alpha, beta) ```

## 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. `mu` This is the location parameter mu. `alpha` This is a mixture parameter alpha, which must be positive. `beta` This is a mixture parameter beta, 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) = (1/(alpha+beta)) exp((theta-mu)/alpha), theta <= mu

• Density 2: p(theta) = (1/(alpha+beta)) exp((mu-theta)/beta), theta > mu

• Inventor: Fieller, et al. (1992)

• Notation 1: theta ~ SL(mu, alpha, beta)

• Notation 2: p(theta) = SL(theta | mu, alpha, beta)

• Parameter 1: location parameter mu

• Parameter 2: mixture parameter alpha > 0

• Parameter 3: mixture parameter beta > 0

• Mean: E(theta) = mu + beta - alpha

• Variance: var(theta) = alpha^2 + beta^2

• Mode: mode(theta) = mu

This is the three-parameter general skew-Laplace distribution, which is an extension of the two-parameter central skew-Laplace distribution. The general form allows the mode to be shifted along the real line with parameter mu. In contrast, the central skew-Laplace has mode zero, and may be reproduced here by setting mu=0.

The general skew-Laplace distribution is a mixture of a negative exponential distribution with mean beta, and the negative of an exponential distribution with mean alpha. The weights of the positive and negative components are proportional to their means. The distribution is symmetric when alpha=beta, in which case the mean is mu.

These functions are similar to those in the `HyperbolicDist` package.

## Value

`dslaplace` gives the density, `pslaplace` gives the distribution function, `qslaplace` gives the quantile function, and `rslaplace` generates random deviates.

## References

Fieller, N.J., Flenley, E.C., and Olbricht, W. (1992). "Statistics of Particle Size Data". Applied Statistics, 41, p. 127–146.

`dalaplace`, `dexp`, `dlaplace`, `dlaplacep`, and `dsdlaplace`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```library(LaplacesDemon) x <- dslaplace(1,0,1,1) x <- pslaplace(1,0,1,1) x <- qslaplace(0.5,0,1,1) x <- rslaplace(100,0,1,1) #Plot Probability Functions x <- seq(from=0.1, to=3, by=0.01) plot(x, dslaplace(x,0,1,1), ylim=c(0,1), type="l", main="Probability Function", ylab="density", col="red") lines(x, dslaplace(x,0,0.5,2), type="l", col="green") lines(x, dslaplace(x,0,2,0.5), type="l", col="blue") legend(1.5, 0.9, expression(paste(mu==0, ", ", alpha==1, ", ", beta==1), paste(mu==0, ", ", alpha==0.5, ", ", beta==2), paste(mu==0, ", ", alpha==2, ", ", beta==0.5)), lty=c(1,1,1), col=c("red","green","blue")) ```