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Wikipedia / 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.

Definition

denote the standard normal probability density function with the cumulative distribution function

where "erf" is the error function

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), is any PDF is any CDF whose PDF is symmetric about zero.

To add location and scale, and that the absolute value of the skewness .

As has been shown,, but a quite accurate (numerical) approximation is:

Estimation

Maximum likelihood. In contrast, the method of moments has a closed-form expression since the skewness equation can be inverted with

.

.

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

Related distributions

The exponentially modified normal distribution. Thus, in terms of the seven states of randomness. 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.

See also

References