The Asymmetric Laplace Distribution

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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

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  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 sqrt(2) / sigma kappa / (1 + κ^2) exp(- sqrt(2) kappa / sigma |x - θ|) if x >= theta, and sqrt(2) / sigma kappa / (1 + κ^2) exp(- sqrt(2) / (sigma kappa) |x - θ|) if x < theta.

The parameter domains of ASL and ASL* are theta in real, sigma > 0, kappa > 0, and mu in real. The relation of mu and kappa are kappa = (sqrt(2 sigma^2 + mu^2) - mu) / sqrt(2 sigma) or mu = sigma / sqrt(2) (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 wccsnow@gmail.com.

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

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## 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)

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