alaplaceUC {VGAM} | R Documentation

## The Laplace Distribution

### Description

Density, distribution function, quantile function and random generation for the 3-parameter asymmetric Laplace distribution with location parameter `location`, scale parameter `scale`, and asymmetry parameter `kappa`.

### Usage

    dalap(x, location=0, scale=1, tau=0.5, kappa=sqrt(tau/(1-tau)),
          log=FALSE)
    palap(q, location=0, scale=1, tau=0.5, kappa=sqrt(tau/(1-tau)))
    qalap(p, location=0, scale=1, tau=0.5, kappa=sqrt(tau/(1-tau)))
    ralap(n, location=0, scale=1, tau=0.5, kappa=sqrt(tau/(1-tau)))

### Arguments

`x, q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1` then the length is taken to be the number required.
`location` | the location parameter xi.
`scale` | the scale parameter sigma. Must consist of positive values.
`tau` | the quantile parameter tau. Must consist of values in (0,1). This argument is used to specify `kappa` and is ignored if `kappa` is assigned.
`kappa` | the asymmetry parameter kappa. Must consist of positive values.
`log` | if `TRUE`, probabilities `p` are given as `log(p)`.

### Details

There are many variants of asymmetric Laplace distributions (ALDs) and this one is known as the ALD by Kotz et al. (2001). See `alaplace3`, the VGAM family function for estimating the three parameters by maximum likelihood estimation, for formulae and details.

### Value

`dalap` gives the density, `palap` gives the distribution function, `qalap` gives the quantile function, and `ralap` generates random deviates.

### Author(s)

T. W. Yee

### References

Kotz, S., Kozubowski, T. J. and Podgorski, K. (2001) The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance, Boston: Birkhauser.

### See Also

`alaplace3`.

### Examples

    x = seq(-5, 5, by=0.01)
    loc = 0; sigma = 1.5; kappa = 2
    ## Not run:
    plot(x, dalap(x, loc, sigma, kappa=kappa), type="l", col="blue",
         main="Blue is density, red is cumulative distribution function",
         ylim=c(0,1), sub="Purple are 5,10,...,95 percentiles",
         las=1, ylab="", cex.main=0.5)
    abline(h=0, col="blue", lty=2)
    lines(qalap(seq(0.05,0.95,by=0.05), loc, sigma, kappa=kappa),
          dalap(qalap(seq(0.05,0.95,by=0.05), loc, sigma, kappa=kappa),
                    loc, sigma, kappa=kappa), col="purple", lty=3, type="h")
    lines(x, palap(x, loc, sigma, kappa=kappa), type="l", col="red")
    abline(h=0, lty=2)
    ## End(Not run)
    palap(qalap(seq(0.05,0.95,by=0.05), loc, sigma, kappa=kappa),
              loc, sigma, kappa=kappa)

[Package VGAM version 0.7-9 Index]