37  Laplace Distribution

The Laplace distribution — also called the double Exponential — consists of two Exponential tails joined at a central point. It is the distribution that minimizes the expected absolute deviation (rather than squared deviation), making it the natural model for data with frequent large deviations and for L1 (least-absolute-deviations) regression.

Formally, the random variate \(X\) defined for all of \(\mathbb{R}\), is said to have a Laplace Distribution (i.e. \(X \sim \text{Laplace}(\mu, b)\)) with location parameter \(\mu \in \mathbb{R}\) and scale parameter \(b > 0\). The Laplace distribution is not built into base R; custom density and distribution functions are used.

37.1 Probability Density Function

\[ f(x) = \frac{1}{2b}\exp\!\left(-\frac{|x-\mu|}{b}\right) \]

The figure below shows examples of the Laplace Probability Density Function for different parameter combinations.

Code 
dlaplace <- function(x, mu, b) {
  exp(-abs(x - mu) / b) / (2 * b)
}

par(mfrow = c(2, 2))
x <- seq(-8, 8, length = 500)

plot(x, dlaplace(x, 0, 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(mu == 0, ",  ", b == 1)))

plot(x, dlaplace(x, 0, 2), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(mu == 0, ",  ", b == 2)))

plot(x, dlaplace(x, 2, 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(mu == 2, ",  ", b == 1)))

plot(x, dlaplace(x, 0, 0.5), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(mu == 0, ",  ", b == 0.5)))

par(mfrow = c(1, 1))
Figure 37.1: Laplace Probability Density Function for various parameter combinations

37.2 Purpose

The Laplace distribution is the double-sided Exponential — it places equal-weight Exponential tails on both sides of the location parameter. Because it minimizes the mean absolute error (MAE) rather than the mean squared error, its MLE corresponds to the sample median rather than the mean. Common applications include:

  • Financial daily returns and log-price changes (heavier tails than Normal)
  • Sparse signal recovery and LASSO regularization (Laplace prior on coefficients)
  • L1 regression (least absolute deviations): residuals assumed Laplace-distributed
  • Image processing: modeling wavelet coefficients which tend to be sparse
  • Robust statistics: less sensitive to outliers than the Normal distribution

Relation to the discrete setting. The Laplace is a double-sided Exponential; since the Exponential is the continuous analog of the Geometric distribution, the Laplace is the continuous analog of the discrete Laplace distribution (also called the bilateral geometric distribution), defined as the difference of two i.i.d. Geometric variates.

37.3 Distribution Function

\[ F(x) = \begin{cases} \dfrac{1}{2}\exp\!\left(\dfrac{x-\mu}{b}\right) & x \leq \mu \\[6pt] 1 - \dfrac{1}{2}\exp\!\left(-\dfrac{x-\mu}{b}\right) & x > \mu \end{cases} \]

The figure below shows the Laplace Distribution Function for \(\mu = 0\) and \(b = 1\).

Code 
plaplace <- function(x, mu, b) {
  ifelse(x <= mu,
         0.5 * exp((x - mu) / b),
         1 - 0.5 * exp(-(x - mu) / b))
}

x <- seq(-6, 6, length = 500)
plot(x, plaplace(x, 0, 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "F(x)", main = "Laplace Distribution Function",
     sub = expression(paste(mu == 0, ",  ", b == 1)))
Figure 37.2: Laplace Distribution Function (location = 0, scale = 1)

37.4 Moment Generating Function

\[ M_X(t) = \frac{e^{\mu t}}{1 - b^2 t^2}, \quad |t| < \frac{1}{b} \]

37.5 1st Uncentered Moment

\[ \mu_1' = \mu \]

37.6 2nd Uncentered Moment

\[ \mu_2' = \mu^2 + 2b^2 \]

37.7 3rd Uncentered Moment

\[ \mu_3' = \mu^3 + 6\mu b^2 \]

37.8 4th Uncentered Moment

\[ \mu_4' = \mu^4 + 12\mu^2 b^2 + 24b^4 \]

37.9 2nd Centered Moment

\[ \mu_2 = 2b^2 \]

37.10 3rd Centered Moment

\[ \mu_3 = 0 \]

The distribution is symmetric about \(\mu\), so all odd centered moments vanish.

37.11 4th Centered Moment

\[ \mu_4 = 24b^4 \]

37.12 Expected Value

\[ \text{E}(X) = \mu \]

37.13 Variance

\[ \text{V}(X) = 2b^2 \]

37.14 Median

\[ \text{Med}(X) = \mu \]

37.15 Mode

\[ \text{Mo}(X) = \mu \]

The density has a sharp peak (cusp) at \(x = \mu\).

37.16 Coefficient of Skewness

\[ g_1 = 0 \]

The distribution is symmetric about \(\mu\).

37.17 Coefficient of Kurtosis

\[ g_2 = \frac{24b^4}{(2b^2)^2} = \frac{24b^4}{4b^4} = 6 \]

The excess kurtosis is \(g_2 - 3 = 3\), indicating substantially heavier tails than the Normal distribution.

37.18 Parameter Estimation

The MLE estimators have closed forms:

\[ \hat\mu = \tilde x \quad \text{(sample median)}, \qquad \hat b = \frac{1}{n}\sum_{i=1}^n |x_i - \hat\mu| \quad \text{(mean absolute deviation)} \]

set.seed(42)
# Simulate Laplace via difference of two Exponentials
mu_true <- 0; b_true <- 2
u <- runif(200)
x_obs <- mu_true + b_true * log(2*ifelse(u <= 0.5, u, 1-u)) * ifelse(u <= 0.5, 1, -1)

# MLE
mu_hat <- median(x_obs)
b_hat  <- mean(abs(x_obs - mu_hat))
cat("MLE mu:", round(mu_hat, 4), "\n")
cat("MLE b: ", round(b_hat, 4), "\n")
cat("True mu:", mu_true, "  True b:", b_true, "\n")
MLE mu: 0.235 
MLE b:  2.1062 
True mu: 0   True b: 2

37.19 R Module

37.19.1 RFC

The Laplace Distribution module is available in RFC under the menu “Distributions / Laplace Distribution”.

37.19.3 R Code

The following code demonstrates Laplace probability calculations:

mu <- 0; b <- 2

dlaplace <- function(x, mu, b) exp(-abs(x - mu) / b) / (2 * b)
plaplace <- function(x, mu, b) {
  ifelse(x <= mu, 0.5 * exp((x - mu)/b), 1 - 0.5 * exp(-(x - mu)/b))
}

# Density at x = 0
dlaplace(0, mu, b)

# P(|X| > 4): both tails beyond 4
1 - plaplace(4, mu, b) + plaplace(-4, mu, b)

# Mean and SD
cat("Mean:", mu, "\n")
cat("SD:", sqrt(2) * b, "\n")
[1] 0.25
[1] 0.1353353
Mean: 0 
SD: 2.828427

37.20 Example

Daily financial log-returns are modeled as \(X \sim \text{Laplace}(\mu = 0, b = 2)\). The probability that the absolute return exceeds 4 is \(e^{-2} \approx 0.135\).

mu <- 0; b <- 2
plaplace <- function(x, mu, b) {
  ifelse(x <= mu, 0.5 * exp((x - mu)/b), 1 - 0.5 * exp(-(x - mu)/b))
}

# P(|return| > 4)
p_tail <- 2 * (1 - plaplace(4, mu, b))
cat("P(|return| > 4):", round(p_tail, 6), "\n")
cat("Exact value exp(-2):", exp(-2), "\n")

# Compare with Normal having same variance
sigma_norm <- sqrt(2) * b
p_norm <- 2 * pnorm(-4, 0, sigma_norm)
cat("Same under Normal:", round(p_norm, 6), "\n")
P(|return| > 4): 0.135335 
Exact value exp(-2): 0.1353353 
Same under Normal: 0.157299
Interactive Shiny app (click to load).
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37.21 Random Number Generator

Laplace random variates are generated via the inverse-CDF method. Setting \(V = U - 1/2\) for \(U \sim \text{U}(0,1)\):

\[ X = \mu - b\,\text{sgn}(V)\ln(1 - 2|V|) \sim \text{Laplace}(\mu, b) \]

set.seed(123)
n <- 1000; mu <- 0; b <- 2

# Inverse-transform method
u <- runif(n)
v <- u - 0.5
x_inv <- mu - b * sign(v) * log(1 - 2*abs(v))

cat("Simulated mean:", round(mean(x_inv), 4), "\n")
cat("Theoretical mean:", mu, "\n")
cat("Simulated var:", round(var(x_inv), 4), "\n")
cat("Theoretical var:", 2 * b^2, "\n")
Simulated mean: -0.032 
Theoretical mean: 0 
Simulated var: 7.8711 
Theoretical var: 8
Interactive Shiny app (click to load).
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37.22 Property 1: Difference of Two Exponentials

If \(Y_1, Y_2 \overset{\text{i.i.d.}}{\sim} \text{Exp}(1)\) then:

\[ X = \mu + b(Y_1 - Y_2) \sim \text{Laplace}(\mu, b) \]

This construction confirms the interpretation as a “double Exponential” and provides an alternative random number generator.

37.23 Property 2: L1 MLE

Maximizing the Laplace log-likelihood is equivalent to minimizing \(\sum|x_i - \mu|\), so the MLE of \(\mu\) is the sample median rather than the mean. This is the basis of L1 (least absolute deviations) regression.

37.24 Property 3: Kurtosis

The kurtosis \(g_2 = 6\) (excess = 3) is substantially greater than the Normal (\(g_2 = 3\)). The Laplace distribution has a sharp central peak and heavier tails, which makes it more sensitive to extreme values than the Logistic (\(g_2 = 4.2\)) but better at capturing impulsive noise than the Normal.