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 $$\gamma_1$$.

External link

 * A very brief introduction to the skew-normal distribution