VGAM (version 1.1-6)

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)), lower.tail = TRUE, log.p = FALSE)
qloglap(p, location.ald = 0, scale.ald = 1, tau = 0.5,
        kappa = sqrt(tau/(1-tau)), lower.tail = TRUE, log.p = FALSE)
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).

lower.tail, log.p

Same meaning as in pnorm or qnorm.

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

# NOT RUN {
loc <- 0; sigma <- exp(0.5); kappa <- 1
x <- seq(-0.2, 5, by = 0.01)
# }
# NOT RUN {
plot(x, dloglap(x, loc, sigma, kappa = kappa),
     type = "l", col = "blue", ylim = c(0,1),
     main = "Blue is density, red is cumulative distribution function",
     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)
# }
# NOT RUN {
ploglap(qloglap(seq(0.05,0.95,by = 0.05), loc, sigma, kappa = kappa),
        loc, sigma, kappa = kappa)
# }