Using the ExtendedLaplace Package

In ExtendedLaplace: The Extended Laplace Distribution

knitr::opts_chunk$set(
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Introduction

The ExtendedLaplace package provides tools for working with the Extended Laplace (EL) distribution, a generalization of the classical Laplace distribution. This distribution is characterized by four parameters: location $\mu$, scale (\sigma > 0), and a uniform noise range (\delta > 0).

The EL distribution arises as the sum (Y = X + U) where (X \sim \text{Laplace}(\mu, \sigma)) and (U \sim \text{Uniform}(-\delta, \delta)).

Installation

To install the development version of this package from GitHub:

# install.packages("devtools")
devtools::install_github("saahdavid/ExtendedLaplace")

Functions

The package provides the following main functions:

• dEL(y, mu, sigma, delta): Probability density function
• pEL(y, mu, sigma, delta): Cumulative distribution function
• qEL(u, mu, sigma, delta): Quantile function (inverse CDF)
• rEL(n, mu, sigma, delta): Random number generation
• qqplotEL(samples, mu, sigma, delta): Quantile-Quantile Plot

Examples

Density

library(ExtendedLaplace)
curve(dEL(x, mu = 0, sigma = 1, delta = 1), from = -5, to = 5, ylab = "Density", xlab = 'y')

Distribution Function

curve(pEL(x, mu = 0, sigma = 1, delta = 1), from = -5, to = 5, ylab = "CDF", xlab = 'y')

Quantiles

qEL(c(0.025, 0.5, 0.975), mu = 0, sigma = 1, delta = 1)

Simulation

samples <- rEL(10000, mu = 0, sigma = 1, delta = 1)
hist(samples, probability = TRUE, breaks = 40, main = "Simulated EL Data", xlab = 'y')
curve(dEL(x, mu = 0, sigma = 1, delta = 1), add = TRUE, col = "navy", lwd = 2)

QQ-Plot

qqplotEL(samples, mu = 0, sigma = 1, delta = 1)

Theoretical Notes

The Extended Laplace distribution has the following form:

PDF

$$ \begin{aligned} g(y) = \frac{1}{4\delta} \begin{cases} e^{\frac{y- \mu + \delta}{\sigma}} - e^{\frac{y- \mu - \delta}{\sigma}}, & y < \mu - \delta \ 2 - e^{-\frac{y - \mu + \delta}{\sigma}} - e^{\frac{y - \mu - \delta}{\sigma}}, & \mu - \delta \leq y < \mu + \delta \ e^{-\frac{y - \mu - \delta}{\sigma}} - e^{-\frac{y - \mu + \delta}{\sigma}}, & y \geq \mu + \delta \end{cases} \end{aligned} $$

CDF

$$ \begin{aligned} G(y) = \frac{1}{4\delta} \begin{cases} \sigma e^{\frac{y- \mu + \delta}{\sigma}} - \sigma e^{\frac{y- \mu - \delta}{\sigma}} , & y < \mu - \delta \ 2(y - \mu + \delta) + \sigma e^{-\frac{y - \mu + \delta}{\sigma}} - \sigma e^{\frac{y - \mu - \delta}{\sigma}} , & \mu-\delta \leq y < \mu+\delta\ 4\delta + \sigma e^{-\frac{y- \mu + \delta}{\sigma}} - \sigma e^{-\frac{y- \mu - \delta}{\sigma}} , & y \geq \mu + \delta \, . \end{cases} \end{aligned} $$

The quantile function

For $\sigma>0$ and $0<u<1$, we have $Q(u)=\mu+\sigma z$, where

$$ \begin{aligned} z = \begin{cases} \log [4\tau u] - \log [e^\tau - e^{-\tau}] & \mbox{for } 0<u\leq (1-e^{-2\tau})/(4\tau) \ z^\ast & \mbox{for } (1-e^{-2\tau})/(4\tau) \leq u \leq 1 - (1-e^{-2\tau})/(4\tau)\ - \log [4\tau (1-u)] + \log [e^{\tau} - e^{-\tau}] & \mbox{for } 1 - (1-e^{-2\tau})/(4\tau) \leq u <1, \end{cases} \end{aligned} $$

where $\tau=\delta/\sigma$ and $z^\ast$ is a unique solution of the equation: $$ \begin{aligned} u = \frac{1}{4\tau} [ 2(z+\tau) - e^{-\tau}(e^z - e^{-z})], \quad -\tau\leq z \leq \tau. \end{aligned} $$

Session Info

sessionInfo()

References

Saah, D. K., & Kozubowski, T. J. (2025).
A new class of extended Laplace distributions with applications to modeling contaminated Laplace data.
Journal of Computational and Applied Mathematics.
https://doi.org/10.1016/j.cam.2025.116588

ExtendedLaplace documentation built on June 8, 2025, 11:10 a.m.