dist.Normal.Laplace {LaplacesDemon} | R Documentation |
Normal-Laplace Distribution: Univariate Asymmetric
Description
These functions provide the density and random generation for the
univariate, asymmetric, normal-Laplace distribution with location
parameter \mu
, scale parameter \sigma
, and
tail-behavior parameters \alpha
and \beta
.
Usage
dnormlaplace(x, mu=0, sigma=1, alpha=1, beta=1, log=FALSE)
rnormlaplace(n, mu=0, sigma=1, alpha=1, beta=1)
Arguments
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 |
sigma |
This is the scale parameter |
alpha |
This is shape parameter |
beta |
This is shape parameter |
log |
Logical. If |
Details
Application: Continuous Univariate
Density:
p(\theta) = \frac{\alpha\beta}{\alpha + \beta}\phi\frac{\theta - \mu}{\sigma} [R(\alpha\sigma - \frac{\theta - \mu}{\sigma}) + R(\beta\sigma + \frac{\theta - \mu}{\sigma})]
Inventor: Reed (2006)
Notation 1:
\theta \sim \mathrm{NL}(\mu,\sigma,\alpha,\beta)
Notation 2:
p(\theta) = \mathrm{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.
Value
dnormlaplace
gives the density, and
rnormlaplace
generates random deviates.
References
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.
See Also
dalaplace
,
dallaplace
,
daml
,
dlaplace
, and
dnorm
Examples
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"))