# alaplaceUC: The Laplace Distribution In VGAM: Vector Generalized Linear and Additive Models

 alaplaceUC 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. 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.

alaplace3, dextlogF, extlogF1.

### 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 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)
## 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 documentation built on Sept. 19, 2023, 9:06 a.m.