dist.Power.Exponential: Power Exponential Distribution: Univariate Symmetric
In LaplacesDemon: Complete Environment for Bayesian Inference

Description Usage Arguments Details Value References See Also Examples

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 mu.
`sigma`	This is the scale parameter sigma, which must be positive.
`kappa`	This is the kurtosis parameter kappa, which must be positive.
`log,log.p`	Logical. If `log=TRUE`, then the logarithm of the density or result is returned.
`lower.tail`	Logical. If `lower.tail=TRUE` (default), probabilities are Pr[X <= x], otherwise, Pr[X > x].

Application: Continuous Univariate
Density: p(theta) = 1/(2 kappa^(1/kappa) Gamma(1+1/kappa) sigma) * exp(-(|theta - mu|^kappa)/(kappa sigma^kappa))
Inventor: Subbotin, M.T. (1923)
Notation 1: theta ~ PE(mu, sigma, kappa)
Notation 2: p(theta) = 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 -> infinity, 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"))