u4_asl: The Asymmetric Laplace Distribution

In cubfits: Codon Usage Bias Fits

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 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 [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)

cubfits documentation built on Nov. 8, 2021, 1:07 a.m.