R: Log-Laplace Distribution: Univariate Symmetric

dist.Log.Laplace {LaplacesDemon} | R Documentation

## Log-Laplace Distribution: Univariate Symmetric

### Description

These functions provide the density, distribution function, quantile function, and random generation for the univariate, symmetric, log-Laplace distribution with location parameter `location` and scale parameter `scale`.

### Usage

    dllaplace(x, location=0, scale=1, log=FALSE)
    pllaplace(q, location=0, scale=1)
    qllaplace(p, location=0, scale=1)
    rllaplace(n, location=0, scale=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.
`log` | Logical. If `log=TRUE`, then the logarithm of the density is returned.

### Details

- Application: Continuous Univariate

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

- Density 2: p(theta) = ((sqrt(2)/lambda)^2 / (2*(sqrt(2)/lambda))) * exp((sqrt(2)/lambda)*(theta - mu)),theta < exp(mu)

- Inventor: Pierre-Simon Laplace

- Notation 1: theta ~ LL(mu, lambda)

- Notation 2: p(theta) = LL(theta | mu, lambda)

- Parameter 1: location parameter mu

- Parameter 2: scale parameter lambda > 0

- Mean: E(theta) =

- Variance: var(theta) =

- Mode: mode(theta) =

The univariate, symmetric log-Laplace distribution is derived from the Laplace distribution. Multivariate and asymmetric versions also exist.

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

### Value

`dllaplace` gives the density, `pllaplace` gives the distribution function, `qllaplace` gives the quantile function, and `rllaplace` generates random deviates.

### References

Kozubowski, T. J. and Podgorski, K. (2003). "Log-Laplace Distributions". International Mathematical Journal, 3, p. 467–495.

### See Also

`dalaplace`, `dallaplace`, `dexp`, `dlaplace`, `dlaplacep`, `dmvl`, `dnorm`, `dnormp`, and `dnormv`.

### Examples

    library(LaplacesDemon)
    x <- dllaplace(1,0,1)
    x <- pllaplace(1,0,1)
    x <- qllaplace(0.5,0,1)
    x <- rllaplace(100,0,1)

    #Plot Probability Functions
    x <- seq(from=0.1, to=20, by=0.1)
    plot(x, dllaplace(x,0,0.1), ylim=c(0,1), type="l", main="Probability Function",
         ylab="density", col="red")
    lines(x, dllaplace(x,0,0.5), type="l", col="green")
    lines(x, dllaplace(x,0,1.5), type="l", col="blue")
    legend(2, 0.9, expression(paste(mu==0, ", ", lambda==0.1),
         paste(mu==0, ", ", lambda==0.5), paste(mu==0, ", ", lambda==1.5)),
         lty=c(1,1,1), col=c("red","green","blue"))

[Package LaplacesDemon version 16.1.1 Index]