R: Asymmetric Log-Laplace Distribution

dist.Asymmetric.Log.Laplace {LaplacesDemon} | R Documentation

## Asymmetric Log-Laplace Distribution

### 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

    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

    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"))

[Package LaplacesDemon version 16.1.1 Index]