Let denote the standard normalprobability density function with the cumulative distribution function given by where "erf" is the error function. Then the probability density function of the skew-normal distribution with parameter is given by This distribution was first introduced by O'Hagan and Leonard. Approximations to this distribution that are easier to manipulate mathematically 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. Both the distribution and its stochastic process underpinnings were consequences of the symmetry argument developed in Chan and Tong, which applies to multivariatecases 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=2 φ Φ where φ is any PDF symmetric about zero and Φ is any CDF whose PDF is symmetric about zero. To add location and scale parameters to this, one makes the usual transform. One can verify that the normal distribution is recovered when, and that the absolute value of the skewness increases as the absolute value of increases. The distribution is right skewed if and is left skewed if. The probability density function with location, scale, and parameter becomes Note, however, that the skewness of the distribution is limited to the interval. As has been shown, the mode of the distribution is unique. For general there's no analytic expression for , but a quite accurate approximation is: where and
Estimation
estimates for,, and can be computed numerically, but no closed-form expression for the estimates is available unless. If a closed-form expression is needed, the method of moments can be applied to estimate from the sample skew, by inverting the skewness equation. This yields the estimate where, and is the sample skew. The sign of is the same as the sign of. Consequently,. The maximum skewness is obtained by setting in the skewness equation, giving. However it is possible that the sample skewness is larger, and then 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 . Concern has been expressed about the impact of skew normal methods on the reliability of inferences based upon them.
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 for some positive. 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. 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 one direction.
