Asymmetric Laplace Distribution {cubfits}

The Asymmetric Laplace Distribution

Description

Density, probability, quantile, random number generation, and MLE functions for the asymmetric Laplace distribution with parameters either in ASL(\theta, \mu, \sigma) or the alternative ASL^*(\theta, \kappa, \sigma).

Usage

  dasl(x, theta = 0, mu = 0, sigma = 1, log = FALSE)
  dasla(x, theta = 0, kappa = 1, sigma = 1, log = FALSE)

  pasl(q, theta = 0, mu = 0, sigma = 1, lower.tail = TRUE,
          log.p = FALSE)
  pasla(q, theta = 0, kappa = 1, sigma = 1, lower.tail = TRUE,
           log.p = FALSE)

  qasl(p, theta = 0, mu = 0, sigma = 1, lower.tail = TRUE,
          log.p = FALSE)
  qasla(p, theta = 0, kappa = 1, sigma = 1, lower.tail = TRUE,
           log.p = FALSE)

  rasl(n, theta = 0, mu = 0, sigma = 1)
  rasla(n, theta = 0, kappa = 1, sigma = 1)

  asl.optim(x)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

theta

center parameter.

mu, kappa

location parameters.

sigma

shape parameter.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X <= x] otherwise, P[X > x].

Details

The density f(x) of ASL^*(\theta, \kappa, \sigma) is given as \frac{\sqrt{2}}{\sigma}\frac{\kappa}{1 + \kappa^2} exp(- \frac{\sqrt{2}\kappa}{\sigma} |x - \theta|) if x \ge \theta, and \frac{\sqrt{2}}{\sigma}\frac{\kappa}{1 + \kappa^2} exp(- \frac{\sqrt{2}}{\sigma\kappa} |x - \theta|) if x < \theta.

The parameter domains of ASL and ASL* are \theta \in R, \sigma > 0, \kappa > 0, and \mu \in R. The relation of \mu and \kappa are \kappa = \frac{\sqrt{2\sigma^2 + \mu^2}-\mu}{\sqrt{2\sigma}} or \mu = \frac{\sigma}{\sqrt{2}} (\frac{1}{\kappa} - \kappa).

Value

“dasl” and “dasla” give the densities, “pasl” and “pasla” give the distribution functions, “qasl” and “qasla” give the quantile functions, and “rasl” and “rasls” give the random numbers.

asl.optim returns the MLE of data x including theta, mu, kappa, and sigma.

Author(s)

Wei-Chen Chen [e-mail].

References

Kotz S, Kozubowski TJ, Podgorski K. (2001) “The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance.” Boston: Birkhauser.

Examples

## Not run: 
suppressMessages(library(cubfits, quietly = TRUE))
set.seed(1234)

dasl(-2:2)
dasla(-2:2)
pasl(-2:2)
pasla(-2:2)
qasl(seq(0, 1, length = 5))
qasla(seq(0, 1, length = 5))

dasl(-2:2, log = TRUE)
dasla(-2:2, log = TRUE)
pasl(-2:2, log.p = TRUE)
pasla(-2:2, log.p = TRUE)
qasl(log(seq(0, 1, length = 5)), log.p = TRUE)
qasla(log(seq(0, 1, length = 5)), log.p = TRUE)

set.seed(123)
rasl(5)
rasla(5)

asl.optim(rasl(5000))

## End(Not run)