dist.Power.Exponential | R Documentation |
These functions provide the density, distribution function, quantile
function, and random generation for the univariate, symmetric, power
exponential distribution with location parameter \mu
, scale
parameter \sigma
, and kurtosis parameter
\kappa
.
dpe(x, mu=0, sigma=1, kappa=2, log=FALSE)
ppe(q, mu=0, sigma=1, kappa=2, lower.tail=TRUE, log.p=FALSE)
qpe(p, mu=0, sigma=1, kappa=2, lower.tail=TRUE, log.p=FALSE)
rpe(n, mu=0, sigma=1, kappa=2)
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 |
sigma |
This is the scale parameter |
kappa |
This is the kurtosis parameter |
log , log.p |
Logical. If |
lower.tail |
Logical. If |
Application: Continuous Univariate
Density: p(\theta) = \frac{1}{2 \kappa^{1/\kappa}
\Gamma(1+\frac{1}{\kappa}) \sigma}
\exp(-\frac{|\theta-\mu|^{\kappa}}{\kappa \sigma^\kappa})
Inventor: Subbotin, M.T. (1923)
Notation 1: \theta \sim \mathcal{PE}(\mu, \sigma, \kappa)
Notation 2: p(\theta) = \mathcal{PE}(\theta | \mu,
\sigma, \kappa)
Parameter 1: location parameter \mu
Parameter 2: scale parameter \sigma > 0
Parameter 3: kurtosis parameter \kappa > 0
Mean: E(\theta) = \mu
Variance: var(\theta) =
Mode: mode(\theta) = \mu
The power exponential distribution is also called the exponential power
distribution, generalized error distribution, generalized Gaussian
distribution, and generalized normal distribution. The original form was
introduced by Subbotin (1923) and re-parameterized by Lunetta
(1963). These functions use the more recent parameterization by Lunetta
(1963). A shape parameter, \kappa > 0
, is added to the
normal distribution. When \kappa=1
, the power exponential
distribution is the same as the Laplace distribution. When
\kappa=2
, the power exponential distribution is the same
as the normal distribution. As \kappa \rightarrow \infty
, this becomes a uniform distribution \in (\mu-\sigma,
\mu+\sigma)
. Tails that are heavier than
normal occur when \kappa < 2
, or lighter than normal
when \kappa > 2
. This distribution is univariate and
symmetric, and there exist multivariate and asymmetric versions.
These functions are similar to those in the normalp
package.
dpe
gives the density,
ppe
gives the distribution function,
qpe
gives the quantile function, and
rpe
generates random deviates.
Lunetta, G. (1963). "Di una Generalizzazione dello Schema della Curva Normale". Annali della Facolt'a di Economia e Commercio di Palermo, 17, p. 237–244.
Subbotin, M.T. (1923). "On the Law of Frequency of Errors". Matematicheskii Sbornik, 31, p. 296–301.
dlaplace
,
dlaplacep
,
dmvpe
,
dnorm
,
dnormp
,
dnormv
, and
dunif
.
library(LaplacesDemon)
x <- dpe(1,0,1,2)
x <- ppe(1,0,1,2)
x <- qpe(0.5,0,1,2)
x <- rpe(100,0,1,2)
#Plot Probability Functions
x <- seq(from=0.1, to=3, by=0.01)
plot(x, dpe(x,0,1,0.1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dpe(x,0,1,2), type="l", col="green")
lines(x, dpe(x,0,1,5), type="l", col="blue")
legend(1.5, 0.9, expression(paste(mu==0, ", ", sigma==1, ", ", kappa==0.1),
paste(mu==0, ", ", sigma==1, ", ", kappa==2),
paste(mu==0, ", ", sigma==1, ", ", kappa==5)),
lty=c(1,1,1), col=c("red","green","blue"))
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