VGAM (version 1.1-9)

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)))

Value

dalap gives the density,

palap gives the distribution function,

qalap gives the quantile function, and

ralap generates random deviates.

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.

Author

T. W. Yee and Kai Huang

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.

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, dextlogF, extlogF1.

Examples

x <- seq(-5, 5, by = 0.01)
loc <- 0; sigma <- 1.5; kappa <- 2
if (FALSE)  plot(x, dalap(x, loc, sigma, kappa = kappa), type = "l",
     main = "Blue is density, orange is the CDF",
     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) 

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