Kurtosis and Departure from Normality

Summary

The coefficients of skewness and kurtosis are traditional measures of departure from normality which have been widely used, particularly in an empirical context. Theoretical disadvantages of the quantities have been often mentioned in the literature; these are here emphasized by an example of a family of non-symmetric distributions, all of whose odd order moments vanish, which have the same moments, the first four coinciding with those of the unit normal law. Nevertheless, if one restricts the class of distributions under consideration to a class L ₂ of mixtures of normals, then the kurtosis appears as a distance in a metric space setting Keilson and Steutel (1974)]. It is shown here that, in this metric space setting, the kurtosis can also be used in a bound on both the uniform metric for the distance between a distribution function and the unit normal distribution function and a non-uniform bound. A similar, and simpler, bound is also given in the case of more general mixtures.