VGAM: alaplaceUC – R documentation

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alaplaceUC

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. The ALD density may be approximated by dextlogF.

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.

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",
     main = "Blue is density, orange is cumulative distr. function",
     ylim = c(0, 1), sub = "Purple are 5, 10, ..., 95 percentiles",
     las = 1, ylab = "", cex.main = 0.5, col = "blue")
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

VGAM

Vector Generalized Linear and Additive Models

v1.1-5

GPL-3

Authors

Thomas Yee [aut, cre], Cleve Moler [ctb] (author of several LINPACK routines)

Initial release

2021-01-13