VGAM (version 0.9-1)

loglapUC: The Log-Laplace Distribution

Description

Density, distribution function, quantile function and random generation for the 3-parameter log-Laplace distribution with location parameter location.ald, scale parameter scale.ald (on the log scale), and asymmetry parameter kappa.

Usage

dloglap(x, location.ald = 0, scale.ald = 1,
        tau = 0.5, kappa = sqrt(tau/(1-tau)), log = FALSE)
ploglap(q, location.ald = 0, scale.ald = 1,
        tau = 0.5, kappa = sqrt(tau/(1-tau)))
qloglap(p, location.ald = 0, scale.ald = 1,
        tau = 0.5, kappa = sqrt(tau/(1-tau)))
rloglap(n, location.ald = 0, scale.ald = 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.ald, scale.ald

the location parameter $\xi$ and the (positive) scale parameter $\sigma$, on the log scale.

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).

Value

• dloglap gives the density, ploglap gives the distribution function, qloglap gives the quantile function, and rloglap generates random deviates.

Details

A positive random variable $Y$ is said to have a log-Laplace distribution if $\log(Y)$ has an asymmetric Laplace distribution (ALD). There are many variants of ALDs and the one used here is described in alaplace3.

References

Kozubowski, T. J. and Podgorski, K. (2003) Log-Laplace distributions. International Mathematical Journal, 3, 467--495.

See Also

dalap, alaplace3, loglaplace1.

Examples

loc <- 0; sigma <- exp(0.5); kappa <- 1
x <- seq(-0.2, 5, by = 0.01)
plot(x, dloglap(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 = "")
abline(h = 0, col = "blue", lty = 2)
lines(qloglap(seq(0.05,0.95,by = 0.05), loc, sigma, kappa = kappa),
      dloglap(qloglap(seq(0.05,0.95,by = 0.05), loc, sigma, kappa = kappa),
                loc, sigma, kappa = kappa), col = "purple", lty = 3, type = "h")
lines(x, ploglap(x, loc, sigma, kappa = kappa), type = "l", col = "red")
abline(h = 0, lty = 2)
ploglap(qloglap(seq(0.05,0.95,by = 0.05), loc, sigma, kappa = kappa),
          loc, sigma, kappa = kappa)