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

## 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

 ```1 2 3 4``` ```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 <= x], otherwise, Pr[X > x].

## Details

• 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.

## 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.

`dlaplace`, `dlaplacep`, `dmvpe`, `dnorm`, `dnormp`, `dnormv`, and `dunif`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```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")) ```