LaplacesDemon (version 16.1.0)

dist.Laplace.Precision: Laplace Distribution: Precision Parameterization

Description

These functions provide the density, distribution function, quantile function, and random generation for the univariate, symmetric, Laplace distribution with location parameter \(\mu\) and precision parameter \(\tau\), which is the inverse of the usual scale parameter, \(\lambda\).

Usage

dlaplacep(x, mu=0, tau=1, log=FALSE)
plaplacep(q, mu=0, tau=1)
qlaplacep(p, mu=0, tau=1)
rlaplacep(n, mu=0, tau=1)

Arguments

x, q

These are each a vector of quantiles.

p

This is a vector of probabilities.

n

This is the number of observations, which must be a positive integer that has length 1.

mu

This is the location parameter \(\mu\).

tau

This is the precision parameter \(\tau\), which must be positive.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Value

dlaplacep gives the density, plaplacep gives the distribution function, qlaplacep gives the quantile function, and rlaplacep generates random deviates.

Details

• Application: Continuous Univariate

• Density: \(p(\theta) = \frac{\tau}{2} \exp(-\tau |\theta-\mu|)\)

• Inventor: Pierre-Simon Laplace (1774)

• Notation 1: \(\theta \sim \mathrm{Laplace}(\mu,\tau^{-1})\)

• Notation 2: \(\theta \sim \mathcal{L}(\mu, \tau^{-1})\)

• Notation 3: \(p(\theta) = \mathrm{Laplace}(\mu,\tau^{-1})\)

• Notation 4: \(p(\theta) = \mathcal{L}(\theta | \mu, \tau^{-1})\)

• Parameter 1: location parameter \(\mu\)

• Parameter 2: precision parameter \(\tau > 0\)

• Mean: \(E(\theta) = \mu\)

• Variance: \(var(\theta) = 2 \tau^{-2}\)

• Mode: \(mode(\theta) = \mu\)

The Laplace distribution is also called the double exponential distribution, because it looks like two exponential distributions back to back with respect to location \(\mu\). It is also called the ``First Law of Laplace'', just as the normal distribution is referred to as the ``Second Law of Laplace''. The Laplace distribution is symmetric with respect to \(\mu\), though there are asymmetric versions of the Laplace distribution. The PDF of the Laplace distribution is reminiscent of the normal distribution; however, whereas the normal distribution is expressed in terms of the squared difference from the mean \(\mu\), the Laplace density is expressed in terms of the absolute difference from the mean, \(\mu\). Consequently, the Laplace distribution has fatter tails than the normal distribution. It has been argued that the Laplace distribution fits most things in nature better than the normal distribution. Elsewhere, there are a large number of extensions to the Laplace distribution, including asymmetric versions and multivariate versions, among many more. These functions provide the precision parameterization for convenience and familiarity in Bayesian inference.

See Also

dalaplace, dexp, dlaplace, dmvl, dnorm, dnormp, and dnormv.

Examples

# NOT RUN {
library(LaplacesDemon)
x <- dlaplacep(1,0,1)
x <- plaplacep(1,0,1)
x <- qlaplacep(0.5,0,1)
x <- rlaplacep(100,0,1)

#Plot Probability Functions
x <- seq(from=-5, to=5, by=0.1)
plot(x, dlaplacep(x,0,0.5), ylim=c(0,1), type="l", main="Probability Function",
     ylab="density", col="red")
lines(x, dlaplacep(x,0,1), type="l", col="green")
lines(x, dlaplacep(x,0,2), type="l", col="blue")
legend(2, 0.9, expression(paste(mu==0, ", ", tau==0.5),
     paste(mu==0, ", ", tau==1), paste(mu==0, ", ", tau==2)),
     lty=c(1,1,1), col=c("red","green","blue"))
# }