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 $$\Gamma(u)=\int_{0}^{+\infty} x^{u-1}\exp\bigl\{-x\bigr\}dx,~u>0.$$
daep(x, alpha, sigma, mu, epsilon, log = FALSE)Computed pdf of AEP distribution at points of vector \(x\).
Vector of observation of requested random realizations.
Tail thickness parameter.
Scale parameter.
Location parameter.
Skewness parameter.
If TRUE, then log\(\bigl(f_{X}(x|\Theta)\bigr)\) is returned.
Mahdi Teimouri
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).
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.
daep(x = 2, alpha = 1.5, sigma = 1, mu = 0, epsilon = 0.5, log = FALSE)
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