LaTex Test (Laplace Distribution)

Inverse CDF Sampling of the Laplace Distribution

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The Laplace distribution has location $\mu \in \mathbb{R}$ and scale $\sigma \in \mathbb{R}^+$ with:

Probability Density Function:

\[f(x \mid \mu, \sigma) = \frac{1}{2\sigma}\text{exp}\left(-\frac{\mid x - \mu \mid}{\sigma}\right), \quad x \in \mathbb{R}\]

Cumulative Density Function:

\[F(x \mid \mu, \sigma) = \frac12 + \frac12 \text{sign}(x-\mu) \left(1-\text{exp}\left(-\frac{\mid x - \mu \mid}{\sigma}\right)\right), \quad x \in \mathbb{R}\]

Inverse CDF:

\[F^{-1}(u \mid \mu, \sigma) = \mu - \sigma \cdot \text{sign}(u - 0.5)\ln(1-2 \mid u - 0.5 \mid), \quad u \in [0,1]\]

LaTex Code:

PDF:
$$f(x \mid \mu, \sigma) = 
    \frac{1}{2\sigma}\text{exp}\left(-\frac{\mid x - \mu \mid}{\sigma}\right), 
    \quad x \in \mathbb{R}$$

CDF:
$$F(x \mid \mu, \sigma) = 
    \frac12 + \frac12 \text{sign}(x-\mu)
    \left(1-\text{exp}\left(-\frac{\mid x - \mu \mid}{\sigma}\right)\right), 
    \quad x \in \mathbb{R}$$

ICDF:
$$F^{-1}(u \mid \mu, \sigma) = 
    \mu - \sigma \cdot \text{sign}(u - 0.5)\ln(1-2 \mid u - 0.5 \mid), 
    \quad u \in [0,1]$$

R Code:

# PDF
dlaplace = function(x, mu, sigma) {
  density = (1/(2*sigma))*exp(-abs(x-mu)/sigma)
  return(density)
}

# CDF
qlaplace = function(u, mu, sigma) {
  FuInv = mu - sigma * sign(u - 0.5) * log(1-2*abs(u - 0.5))
  return(FuInv)
}

# ICDF sampling
set.seed(1)
U = runif(1000)
mu = 1; sigma = 2
X = qlaplace(U, mu, sigma)

# Empirical vs. Theoretical density
xx = seq(-15, 15, 0.1)
fx = dlaplace(xx, mu, sigma)
hist(X, 
     main = "Empirical vs. Theoretical density", 
     ylim = c(0, max(fx)), 
     col = rgb(232, 74, 39, max = 255, alpha = 200),
     freq = F)
lines(xx, fx, type = 'l', lwd = 2, col = '#13294b')

This is a plot comparing the empirical (histogram) and theoretical (curve) densities of the Laplace distribution.

Liquid Code:

{ %
  include figure image_path="/assets/images/mathstat-posts/laplace_EvT_densityplot.png"
  alt="This is a plot comparing the empirical (histogram) and theoretical (curve) densities of the Laplace distribution."
  caption="This is a plot comparing the empirical (histogram) and theoretical (curve) densities of the Laplace distribution."
% }