The AEP distribution with alpha = 0.5 (fixed for identifiability).
aep_cdf(x, theta1, theta2)CDF values
Numeric vector
Left tail parameter (theta1 > 0)
Right tail parameter (theta2 > 0)
For x <= 0: F(x) = 0.5 * (1 - P(1/theta1, u1)) where u1 = (|x| * 2 * Gamma(1 + 1/theta1))^theta1
For x > 0: F(x) = 0.5 + 0.5 * P(1/theta2, u2) where u2 = (x * 2 * Gamma(1 + 1/theta2))^theta2
P(a, x) is the regularized incomplete gamma function (pgamma).
Special case: theta1 = theta2 gives a symmetric distribution. Note: theta = 2 has a Gaussian kernel but is NOT equal to probit due to scaling.
Reference: Naranjo et al. (2015) Statistics and Computing