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. Let formula_12 denote the standard normal probability density function with the cumulative distribution function given by where "erf" is the error function. Then the probability density function (pdf) of the skew-normal distribution with parameter formula_15 is given by 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 formula_17 where formula_18 is any PDF symmetric about zero and formula_19 is any CDF whose PDF is symmetric about zero. To add location and scale parameters to this, one makes the  usual transform formula_20. One can verify that the normal distribution is recovered when formula_21, and that the absolute value of the skewness increases as the absolute value of formula_15 increases. The distribution is right skewed if formula_23 and is left skewed if formula_24. The probability density function with location formula_25, scale formula_26, and parameter formula_15 becomes Note, however, that the skewness (formula_29) of the distribution is limited to the interval formula_30. As has been shown, the mode (maximum) of the distribution is unique. For general formula_31 there's no analytic expression for formula_32 , but a quite accurate (numerical) approximation is : where formula_34 and formula_35 Estimation. Maximum likelihood estimates for formula_25, formula_26, and formula_15 can be computed numerically, but no closed-form expression for the estimates is available unless formula_39. If a closed-form expression is needed, the method of moments can be applied to estimate formula_15 from the sample skew, by inverting the skewness equation. This yields the estimate 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. Skew normal distribution