GeneralizedHyperbolic (version 0.8-4)

SkewLaplace: Skew-Laplace Distribution

Description

Density function, distribution function, quantiles and random number generation for the skew-Laplace distribution.

Usage

dskewlap(x, mu = 0, alpha = 1, beta = 1,
         param = c(mu, alpha, beta), logPars = FALSE)
pskewlap(q, mu = 0, alpha = 1, beta = 1,
         param = c(mu, alpha, beta))
qskewlap(p, mu = 0, alpha = 1, beta = 1,
         param = c(mu, alpha, beta))
rskewlap(n, mu = 0, alpha = 1, beta = 1,
         param = c(mu, alpha, beta))

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations to be generated.

mu

The location parameter, set to 0 by default.

alpha, beta

The shape parameters, both set to 1 by default.

param

Vector of parameters of the skew-Laplace distribution: \(\mu\), \(\alpha\) and \(\beta\)

logPars

Logical. If TRUE the second and third components of param are taken to be \(log(\alpha)\) and \(log(\beta)\) respectively.

Value

dskewlap gives the density, pskewlap gives the distribution function, qskewlap gives the quantile function and rskewlap generates random variates. The distribution function is obtained by elementary integration of the density function. Random variates are generated from exponential observations using the characterization of the skew-Laplace as a mixture of exponential observations.

Details

The central skew-Laplace has mode zero, and is a mixture of a (negative) exponential distribution with mean \(\beta\), and the negative of an exponential distribution with mean \(\alpha\). The weights of the positive and negative components are proportional to their means.

The general skew-Laplace distribution is a shifted central skew-Laplace distribution, where the mode is given by \(\mu\).

The density is given by: $$f(x)=\frac{1}{\alpha+\beta} e^{(x - \mu)/\alpha}$$ for \(x\leq\mu\), and $$f(x)=\frac{1}{\alpha+\beta} e^{-(x - \mu)/\beta}$$ for \(x\geq\mu\)

References

Fieller, N. J., Flenley, E. C. and Olbricht, W. (1992) Statistics of particle size data. Appl. Statist., 41, 127--146.

See Also

hyperbFitStart

Examples

# NOT RUN {
param <- c(1, 1, 2)
par(mfrow = c(1, 2))
curve(dskewlap(x, param = param), from = -5, to = 8, n = 1000)
title("Density of the\n Skew-Laplace Distribution")
curve(pskewlap(x, param = param), from = -5, to = 8, n = 1000)
title("Distribution Function of the\n Skew-Laplace Distribution")
dataVector <- rskewlap(500, param = param)
curve(dskewlap(x, param = param), range(dataVector)[1], range(dataVector)[2],
      n = 500)
hist(dataVector, freq = FALSE, add = TRUE)
title("Density and Histogram\n of the Skew-Laplace Distribution")
DistributionUtils::logHist(dataVector, main =
        "Log-Density and Log-Histogram\n of the Skew-Laplace Distribution")
curve(log(dskewlap(x, param = param)), add = TRUE,
      range(dataVector)[1], range(dataVector)[2], n = 500)
# }