# 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 $\phi (x)$ denote the standard normal distribution function

$\phi (x)=N(0,1)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}$ with the cumulative distribution function (CDF) given by

$\Phi (x)=\int _{-\infty }^{x}\phi (t)\ dt={\frac {1}{2}}\left[1+{\text{erf}}\left({\frac {x}{\sqrt {2}}}\right)\right]$ Then the equivalent skew-normal distribution is given by

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

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

$f(x)={\frac {2}{\omega {\sqrt {2\pi }}}}e^{-{\frac {(x-\xi )^{2}}{2\omega ^{2}}}}\int _{-\infty }^{\alpha \left({\frac {x-\xi }{\omega }}\right)}{\frac {1}{\sqrt {2\pi }}}e^{-{\frac {t^{2}}{2}}}\ dt$ One can verify that the normal distribution is recovered in the limit $\alpha \rightarrow 0$ , and that the absolute value of the skewness increases as the absolute value of $\alpha$ increases.

### Moments

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

mean = $\mu =\xi +\omega \delta {\sqrt {\frac {2}{\pi }}}$ variance = $\sigma ^{2}=\omega ^{2}\left(1-{\frac {2\delta ^{2}}{\pi }}\right)$ skewness = $\gamma _{1}={\frac {4-\pi }{2}}{\frac {\left(\delta {\sqrt {2/\pi }}\right)^{3}}{\left(1-2\delta ^{2}/\pi \right)^{3/2}}}$ kurtosis = $2(\pi -3){\frac {\left(\delta {\sqrt {2/\pi }}\right)^{4}}{\left(1-2\delta ^{2}/\pi \right)^{2}}}$ Generally one wants to estimate the distribution's parameters from the standard mean, variance and skewness. The skewness equation can be inverted. This yields

$|\delta |={\sqrt {\frac {\pi }{2}}}{\frac {|\gamma _{1}|^{\frac {1}{3}}}{\sqrt {\gamma _{1}^{\frac {2}{3}}+((4-\pi )/2)^{\frac {2}{3}}}}}$ The sign of $\delta$ is the same as that of $\gamma _{1}$ . 