dist.Power.Exponential {LaplacesDemon}

Power Exponential Distribution: Univariate Symmetric

Description

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.

Usage

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)

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.

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 \le x], otherwise, Pr[X > x].

Details

• 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.

Value

dpe gives the density, ppe gives the distribution function, qpe gives the quantile function, and rpe generates random deviates.

References

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.

See Also

dlaplace, dlaplacep, dmvpe, dnorm, dnormp, dnormv, and dunif.

Examples

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"))