Skew normal distribution

Definition

Let \phi(x) denote the standard normal probability density function :\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} with the cumulative distribution function given by :\Phi(x) = \int_{-\infty}^{x} \phi(t)\ \mathrm dt = \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{x}{\sqrt{2}}\right)\right],

where "erf" is the error function. Then the probability density function (pdf) of the skew-normal distribution with parameter \alpha is given by :f(x) = 2\phi(x)\Phi(\alpha x). ,

This distribution was first introduced by O'Hagan and Leonard (1976). Alternative forms to this distribution, with the corresponding quantile function, have been given by Ashour and Abdel-Hamid and by Mudholkar and Hutson.

A stochastic process that underpins the distribution was described by Andel, Netuka and Zvara (1984). Both the distribution and its stochastic process underpinnings were consequences of the symmetry argument developed in Chan and Tong (1986), which applies to multivariate cases beyond normality, e.g. skew multivariate t distribution and others. The distribution is a particular case of a general class of distributions with probability density functions of the form f(x) = 2 \phi(x) \Phi(x) where \phi(\cdot) is any PDF symmetric about zero and \Phi(\cdot) is any CDF whose PDF is symmetric about zero.

To add location and scale parameters to this, one makes the usual transform x\rightarrow\frac{x-\xi}{\omega}. One can verify that the normal distribution is recovered when \alpha = 0, and that the absolute value of the skewness increases as the absolute value of \alpha increases. The distribution is right skewed if \alpha0 and is left skewed if \alpha. The probability density function with location \xi, scale \omega, and parameter \alpha becomes :f(x) = \frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\Phi\left(\alpha \left(\frac{x-\xi}{\omega}\right)\right). , The skewness ( \gamma_1 ) of the distribution is limited to slightly less than the interval (-1,1) .

As has been shown, the mode (maximum) m_o of the distribution is unique. For general \alpha there is no analytic expression for m_o , but a quite accurate (numerical) approximation is:

\begin{align} \delta &= \frac{\alpha}{\sqrt{1+\alpha^2}} \ m_o (\alpha) &\approx \sqrt{\frac{2}{\pi}}\delta - \left(1-\frac{\pi}{4}\right) \frac{\left(\sqrt{\frac{2}{\pi}}\delta\right)^3}{1-\frac{2}{\pi}\delta^2} - \frac{\mathrm{sgn}(\alpha)}{2} e^{\left(-\frac{2 \pi}{|\alpha |}\right)} \ \end{align}

Estimation

Maximum likelihood estimates for \xi, \omega, and \alpha can be computed numerically, but no closed-form expression for the estimates is available unless \alpha=0. In contrast, the method of moments has a closed-form expression since the skewness equation can be inverted with

:|\delta| = \sqrt{\frac{\pi}{2} \frac{ |\gamma_1|^{\frac{2}{3}} }{|\gamma_1|^{\frac{2}{3}}+((4-\pi)/2)^\frac{2}{3}}}

where \delta = \frac{\alpha}{\sqrt{1+\alpha^2}} and the sign of \delta is the same as the sign of \gamma_1. Consequently, \alpha = \frac{\delta}{\sqrt{1-\delta^2}}, \omega = \frac{\sigma}{\sqrt{1-2\delta^2/\pi}}, and \xi=\mu-\omega\delta\sqrt{\frac{2}{\pi}} where \mu and \sigma are the mean and standard deviation. As long as the sample skewness \hat{\gamma}_1 is not too large, these formulas provide method of moments estimates \hat\alpha, \hat\omega, and \hat\xi based on a sample's \hat\mu, \hat\sigma, and \hat\gamma_1.

The maximum (theoretical) skewness is obtained by setting {\delta = 1} in the skewness equation, giving \gamma_1 \approx 0.9952717. However it is possible that the sample skewness is larger, and then \alpha cannot be determined from these equations. When using the method of moments in an automatic fashion, for example to give starting values for maximum likelihood iteration, one should therefore let (for example) |\hat{\gamma}_1| = \min(0.99, |(1/n)\sum{((x_i-\hat\mu)/\hat\sigma)^3}|).

Concern has been expressed about the inference of skew normal distributions using the direct parameterization.

Related distributions

The exponentially modified normal distribution is another 3-parameter distribution that is a generalization of the normal distribution to skewed cases. The skew normal still has a normal-like tail in the direction of the skew, with a shorter tail in the other direction; that is, its density is asymptotically proportional to e^{-kx^2} for some positive k. Thus, in terms of the seven states of randomness, it shows "proper mild randomness". In contrast, the exponentially modified normal has an exponential tail in the direction of the skew; its density is asymptotically proportional to e^{-k|x|}. In the same terms, it shows "borderline mild randomness".

Thus, the skew normal is useful for modeling skewed distributions which nevertheless have no more outliers than the normal, while the exponentially modified normal is useful for cases with an increased incidence of outliers in (just) one direction.

References

References

1. (1976). "Bayes estimation subject to uncertainty about parameter constraints". Biometrika.
2. (October 2010). "Approximate skew normal distribution". Journal of Advanced Research.
3. (February 2000). "The epsilon–skew–normal distribution for analyzing near-normal data". Journal of Statistical Planning and Inference.
4. [http://dml.cz/bitstream/handle/10338.dmlcz/124493/Kybernetika_20-1984-2_1.pdf Andel, J., Netuka, I. and Zvara, K. (1984) On threshold autoregressive processes. Kybernetika, 20, 89-106]
5. (March 1986). "A note on certain integral equations associated with non-linear time series analysis". Probability Theory and Related Fields.
6. (2000). "Problems of inference for Azzalini's skewnormal distribution". Journal of Applied Statistics.