Skew normal distribution

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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.

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[edit] Definition

Let φ(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 α.

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 α increases.

[edit] 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 δ is the same as that of γ1.

[edit] See also

[edit] Reference

  • Azzalini, A. (1985). "A class of distributions which includes the normal ones". Scand. J. Statist. 12: 171-178. 

[edit] External link

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Probability distributions
 
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