dist.Normal.Laplace: Normal-Laplace Distribution: Univariate Asymmetric
In LaplacesDemon: Complete Environment for Bayesian Inference

Description Usage Arguments Details Value References See Also Examples

These functions provide the density and random generation for the univariate, asymmetric, normal-Laplace distribution with location parameter mu, scale parameter σ, and tail-behavior parameters alpha and beta.

1 2	dnormlaplace(x, mu=0, sigma=1, alpha=1, beta=1, log=FALSE) rnormlaplace(n, mu=0, sigma=1, alpha=1, beta=1)

`x`	This is a vector of data.
`n`	This is the number of observations, which must be a positive integer that has length 1.
`mu`	This is the location parameter mu.
`sigma`	This is the scale parameter sigma, which must be positive.
`alpha`	This is shape parameter alpha for left-tail behavior.
`beta`	This is shape parameter beta for right-tail behavior.
`log`	Logical. If `log=TRUE`, then the logarithm of the density is returned.

Application: Continuous Univariate
Density: p(theta) = [(alpha*beta)/(alpha + beta)] phi [(theta - mu) / sigma] [R(alpha*sigma - [(theta - mu) / sigma]) + R(beta*sigma + [(theta - mu) / sigma])]
Inventor: Reed (2006)
Notation 1: theta ~ NL(mu, sigma, alpha, beta)
Notation 2: p(theta) = NL(theta | mu, sigma, alpha, beta)
Parameter 1: location parameter mu
Parameter 2: scale parameter sigma > 0
Parameter 3: shape parameter alpha > 0
Parameter 4: shape parameter beta > 0
Mean:
Variance:
Mode:

The normal-Laplace (NL) distribution is the convolution of a normal distribution and a skew-Laplace distribution. When the NL distribution is symmetric (when alpha = beta), it behaves somewhat like the normal distribution in the middle of its range, somewhat like the Laplace distribution in its tails, and functions generally between the normal and Laplace distributions. Skewness is parameterized by including a skew-Laplace component. It may be applied, for example, to the logarithmic price of a financial instrument.

Parameters alpha and beta determine the behavior in the left and right tails, respectively. A small value corresponds to heaviness in the corresponding tail. As sigma approaches zero, the NL distribution approaches a skew-Laplace distribution. As beta approaches infinity, the NL distribution approaches a normal distribution, though it never quite reaches it.

dnormlaplace gives the density, and rnormlaplace generates random deviates.

Reed, W.J. (2006). "The Normal-Laplace Distribution and Its Relatives". In Advances in Distribution Theory, Order Statistics and Inference, p. 61–74, Birkhauser, Boston.

dalaplace, dallaplace, daml, dlaplace, and dnorm

library(LaplacesDemon)
x <- dnormlaplace(1,0,1,0.5,2)
x <- rnormlaplace(100,0,1,0.5,2)

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