Density-Based Skewness and Kurtosis Functions

New, functional, concepts of skewness and kurtosis are introduced for large classes of continuous univariate distributions. They are the first skewness and kurtosis measures to be defined directly in terms of the probability density function and its derivative, and are directly interpretable in terms of them. Unimodality of the density is a basic prerequisite. The mode defines the centre of such densities, separating their left and right parts. Skewness is then simply defined by suitably invariant comparison of the distances to the right and left of the mode at which the density is the same, positive function values arising when the former distance is larger. Our skewness functions are, thus, directly interpretable right-left comparisons which characterise asymmetry, vanishing only in the symmetric case. Kurtosis is conceived separately for the left and right parts of a unimodal density, these concepts coinciding in the symmetric case. By reflection in the mode, it suffices to consider right kurtosis. This, in turn, is directly and straightforwardly defined as skewness of an appropriate unimodal function of the right density derivative, two alternative functions being of particular interest. Dividing the right density into its peak and tail parts at the mode of such a function, (right) kurtosis is seen as a corresponding tail-peak comparison. A number of properties and illustrations of both skewness and kurtosis functions are presented and a concept of relative kurtosis addressed. Estimation of skewness and kurtosis functions, via kernel density estimation, is briefly considered and illustrated. Scalar summary skewness and kurtosis measures based on suitable averages of their functional counterparts are also considered and a link made to a popular existing scalar skewness measure. Further developments are briefly