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)),
          lower.tail = TRUE, log.p = FALSE)
    qalap(p, location = 0, scale = 1, tau = 0.5, kappa = sqrt(tau/(1-tau)),
          lower.tail = TRUE, log.p = FALSE)
    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)`.
`lower.tail, log.p` | Same meaning as in `pnorm` or `qnorm`.

### 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 and Kai Huang

### 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, orange 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 = "orange")
    abline(h = 0, lty = 2)
    ## End(Not run)

    pp <- seq(0.05, 0.95, by = 0.05)  # Test two functions
    max(abs(palap(qalap(pp, loc, sigma, kappa = kappa),
                  loc, sigma, kappa = kappa) - pp))  # Should be 0

[Package VGAM version 1.1-1 Index]