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]