daep {AEP}

Computing the probability density function (pdf) of asymmetric exponential power (AEP) distribution.

Description

The pdf of AEP distribution given by

f_{X}(x|\Theta)= \frac{1}{2\sigma \Gamma\bigl(1+\frac{1}{\alpha}\bigr)}\exp\biggl\{-\bigg|\frac{\mu-x}{\sigma(1-\epsilon)}\bigg|^{\alpha}\biggr\},~~{}~x < \mu,

f_{X}(x|\Theta)= \frac{1}{2\sigma \Gamma\bigl(1+\frac{1}{\alpha}\bigr)}\exp\biggl\{-\bigg|\frac{x-\mu}{\sigma(1+\epsilon)}\bigg|^{\alpha}\biggr\},~~{}~x \geq\mu,

where -\infty<x<+\infty, \Theta=(\alpha,\sigma,\mu,\epsilon)^T with 0<\alpha \leq 2, \sigma> 0, -\infty<\mu<\infty, -1<\epsilon<1, and

\Gamma(u)=\int_{0}^{+\infty} x^{u-1}\exp\bigl\{-x\bigr\}dx,~u>0.

Usage

daep(x, alpha, sigma, mu, epsilon, log = FALSE)

Arguments

x

Vector of observation of requested random realizations.

alpha

Tail thickness parameter.

sigma

Scale parameter.

mu

Location parameter.

epsilon

Skewness parameter.

log

If TRUE, then log\bigl(f_{X}(x|\Theta)\bigr) is returned.

Details

The AEP distribution is a special case of asymmetric exponential power distribution studied by Dongming and Zinde-Walsh (2009) when p_1=p_2=\alpha. Also, note that if \epsilon=0, then the AEP distribution turns into a normal distribution with mean \mu and standard deviation \sqrt{2}\sigma. When \alpha=2, the AEP distribution is a slight variant of epsilon-skew-normal distribution introduced by Mudholkar and Huston (2001).

Value

Computed pdf of AEP distribution at points of vector x.

Author(s)

Mahdi Teimouri

References

Z. Dongming and V. Zinde-Walsh, 2009. Properties and estimation of asymmetric exponential power distribution, Journal of Econometrics, 148(1), 86-99.

G. S. Mudholkar and A. D. Huston, 2001. The epsilon-skew-normal distribution for analyzing near-normal data, Journal of Statistical Planning and Inference, 83, 291-309.

Examples

daep(x = 2, alpha = 1.5, sigma = 1, mu = 0, epsilon = 0.5, log = FALSE)