# Skew normal distribution

In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

The normal distribution is a highly important distribution in statistics. In particular, as a result of the central limit theorem, many real-world phenomena are well described by a normal distribution, even if the underlying generative process is not known.

Nevertheless, many phenomena may approach the normal distribution only in the limit of a very large number of events. Other distributions, may never approach the normal distribution due to inherent biases in the underlying process. As a result of this, even with a reasonable number of measurements, a distribution may retain a significant non-zero skewness. The normal distribution cannot be used to model such a distribution as its third order moment (its skewness) is zero. The skew normal distribution is a simple parametric approach to distributions which deviate from the normal distribution only substantially in their skewness. A parametric approximation to the distribution may link the parameters to underlying processes. Furthermore, the existence of a parametric form readily aids hypothesis testing.

## Definition

Let ${\displaystyle \phi (x)}$ denote the standard normal distribution function

${\displaystyle \phi (x)=N(0,1)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}}$

with the cumulative distribution function (CDF) given by

${\displaystyle \Phi (x)=\int _{-\infty }^{x}\phi (t)\ dt={\frac {1}{2}}\left[1+{\text{erf}}\left({\frac {x}{\sqrt {2}}}\right)\right]}$

Then the equivalent skew-normal distribution is given by

${\displaystyle f(x)=2\phi (x)\Phi (\alpha x)\,}$ for some parameter ${\displaystyle \alpha }$.

To add location and scale parameters to this (corresponding to mean and standard deviation for the normal distribution), one makes the usual transform ${\displaystyle x\rightarrow {\frac {x-\xi }{\omega }}}$. This yields the general skew-normal distribution function

${\displaystyle f(x)={\frac {2}{\omega {\sqrt {2\pi }}}}e^{-{\frac {(x-\xi )^{2}}{2\omega ^{2}}}}\int _{-\infty }^{\alpha \left({\frac {x-\xi }{\omega }}\right)}{\frac {1}{\sqrt {2\pi }}}e^{-{\frac {t^{2}}{2}}}\ dt}$

One can verify that the normal distribution is recovered in the limit ${\displaystyle \alpha \rightarrow 0}$, and that the absolute value of the skewness increases as the absolute value of ${\displaystyle \alpha }$ increases.

### Moments

Define ${\displaystyle \delta ={\frac {\alpha }{\sqrt {1+\alpha ^{2}}}}}$. Then we have:

mean = ${\displaystyle \mu =\xi +\omega \delta {\sqrt {\frac {2}{\pi }}}}$
variance = ${\displaystyle \sigma ^{2}=\omega ^{2}\left(1-{\frac {2\delta ^{2}}{\pi }}\right)}$
skewness = ${\displaystyle \gamma _{1}={\frac {4-\pi }{2}}{\frac {\left(\delta {\sqrt {2/\pi }}\right)^{3}}{\left(1-2\delta ^{2}/\pi \right)^{3/2}}}}$
kurtosis = ${\displaystyle 2(\pi -3){\frac {\left(\delta {\sqrt {2/\pi }}\right)^{4}}{\left(1-2\delta ^{2}/\pi \right)^{2}}}}$

Generally one wants to estimate the distribution's parameters from the standard mean, variance and skewness. The skewness equation can be inverted. This yields

${\displaystyle |\delta |={\sqrt {\frac {\pi }{2}}}{\frac {|\gamma _{1}|^{\frac {1}{3}}}{\sqrt {\gamma _{1}^{\frac {2}{3}}+((4-\pi )/2)^{\frac {2}{3}}}}}}$

The sign of ${\displaystyle \delta }$ is the same as that of ${\displaystyle \gamma _{1}}$.