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:$phi(x)=N(0,1)=frac\{1\}\{sqrt\{2pie^\{-frac\{x^2\}\{2$with the cumulative distribution function (CDF) given by:$Phi(x)\; =\; int\_\{-infty\}^\{x\}\; phi(t)\; dt\; =\; frac\{1\}\{2\}\; left\; [\; 1\; +\; ext\{erf\}\; left(frac\{x\}\{sqrt\{2\; ight)\; ight]$

Then the equivalent skew-normal distribution is given by:$f(x)\; =\; 2phi(x)Phi(alpha\; x)\; ,$ 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 $x\; ightarrowfrac\{x-xi\}\{omega\}$. This yields the general skew-normal distribution function:$f(x)\; =\; frac\{2\}\{omegasqrt\{2pi\; e^\{-frac\{(x-xi)^2\}\{2omega^2\; int\_\{-infty\}^\{alphaleft(frac\{x-xi\}\{omega\}\; ight)\}\; frac\{1\}\{sqrt\{2pi\; e^\{-frac\{t^2\}\{2\; dt$

One can verify that the normal distribution is recovered in the limit $alpha\; ightarrow\; 0$, 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 = $mu\; =\; xi\; +\; omegadeltasqrt\{frac\{2\}\{pi$:variance = $sigma^2\; =\; omega^2left(1\; -\; frac\{2delta^2\}\{pi\}\; ight)$:skewness = $gamma\_1\; =\; frac\{4-pi\}\{2\}\; frac\{left(deltasqrt\{2/pi\}\; ight)^3\}\{\; left(1-2delta^2/pi\; ight)^\{3/2\}\; \}$:kurtosis = $2(pi\; -\; 3)frac\{left(deltasqrt\{2/pi\}\; ight)^4\}\{left(1-2delta^2/pi\; ight)^2\}$

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 $gamma\_1$.

ee also

* Normal distribution
* Skewness

References

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External links

* [http://azzalini.stat.unipd.it/SN/Intro/intro.html A very brief introduction to the skew-normal distribution]