15.4 Skew Normal Distribution

15.4.1 Probability Density Function

If \(\xi \in \mathbb{R}\), \(\omega \in \mathbb{R}^+\), and \(\alpha \in \mathbb{R}\), then for \(y \in \mathbb{R}\), \[ \text{SkewNormal}(y \mid \xi, \omega, \alpha) = \frac{1}{\omega\sqrt{2\pi}} \ \exp\left( - \, \frac{1}{2} \left( \frac{y - \xi}{\omega} \right)^2 \right) \ \left(1 + \text{erf}\left( \alpha\left(\frac{y - \xi}{\omega\sqrt{2}}\right)\right)\right) . \]

15.4.2 Sampling Statement

y ~ skew_normal(xi, omega, alpha)

Increment target log probability density with skew_normal_lpdf( y | xi, omega, alpha) dropping constant additive terms.

15.4.3 Stan Functions

real skew_normal_lpdf(reals y | reals xi, reals omega, reals alpha)
The log of the skew normal density of y given location xi, scale omega, and shape alpha

real skew_normal_cdf(reals y, reals xi, reals omega, reals alpha)
The skew normal distribution function of y given location xi, scale omega, and shape alpha

real skew_normal_lcdf(reals y | reals xi, reals omega, reals alpha)
The log of the skew normal cumulative distribution function of y given location xi, scale omega, and shape alpha

real skew_normal_lccdf(reals y | reals xi, reals omega, reals alpha)
The log of the skew normal complementary cumulative distribution function of y given location xi, scale omega, and shape alpha

R skew_normal_rng(reals xi, reals omega, real alpha)
Generate a skew normal variate with location xi, scale omega, and shape alpha; may only be used in generated quantities block. For a description of argument and return types, see section vectorized PRNG functions.