Skew-Normal Distribution#

Univariate, Continuous, Asymmetric, Unbounded, Light-tailed

The skew-normal distribution is a continuous probability distribution that generalizes the normal distribution by introducing a skewness parameter (\(\alpha\)). This parameter allows the distribution to be asymmetric around its mean, with \(\alpha\) determining the direction and degree of the asymmetry. When \(\alpha = 0\), the skew-normal distribution reduces to the normal distribution.

The skew-normal distribution is often used to model data that exhibit skewness, such as financial returns, income distributions, and reaction times. In these cases, the skew-normal distribution provides a flexible framework to capture the asymmetry in the data, which is not possible with the normal distribution.

Key Properties and Parameters#

Support

\(x \in \mathbb{R}\)

Mean

\(\mu + \sigma \sqrt{\frac{2}{\pi}} \frac{\alpha }{{\sqrt {1+\alpha ^{2}}}}\)

Variance

\(\sigma^2 \left( 1-\frac{2\alpha^2}{(\alpha^2+1) \pi} \right)\)

Parameters:

  • \(\mu\) : (float) Location parameter.

  • \(\sigma\) : (float) Scale parameter.

  • \(\tau\) : (float) Precision parameter.

  • \(\alpha\) : (float) Skewness parameter.

Alternative parametrization

The skew-normal distribution has 2 alternative parameterizations. In terms of \(\mu\), \(\sigma\) and \(\alpha\), or in terms of \(\mu\), \(\tau\) and \(\alpha\). Where \(\mu\) is the location parameter, \(\sigma\) is the scale parameter, \(\tau\) is the precision parameter, and \(\alpha\) is the skewness parameter.

The link between the 2 alternatives is given by:

\[ \tau = \frac{1}{\sigma^2} \]

Probability Density Function (PDF)#

\[ f(x \mid \mu, \tau, \alpha) = 2 \Phi((x-\mu)\sqrt{\tau}\alpha) \phi(x,\mu,\tau) \]

where \(\Phi\) is the standard normal CDF and \(\phi\) is the normal PDF.

../../_images/skewnormal_0_1.png
../../_images/skewnormal_1_0.png

Cumulative Distribution Function (CDF)#

\[ F(x \mid \mu, \sigma, \alpha) = \frac{1}{2} \left( 1 + \text{erf} \left( \frac{x - \mu}{\sigma \sqrt{2}} \right) \right) - 2 T \left( \frac{x - \mu}{\sigma}, \alpha \right) \]

where \(\text{erf}\) is the error function and \(T\) is the Owen’s T function.

../../_images/skewnormal_2_0.png
../../_images/skewnormal_3_0.png

See also

Related Distributions:

  • Normal Distribution - The parent distribution from which the skew-normal distribution is derived. When \(\alpha = 0\), the skew-normal distribution reduces to the normal distribution.

  • Half-Normal Distribution - When \(\alpha\) approaches +/- infinity, the skew-normal distribution becomes a half-normal distribution.

References#