Shape measures based on the convex transform order

Three functional measures of the shape of univariate distributions are proposed which are consistent with respect to the convex transform order. The first two are weighted tail indices that characterize location-scale families of distributions, whilst the third is a skewness measure. Properties of the new measures are established for various classes of symmetric and asymmetric distributions, and the generalized Pareto distribution characterized in terms of them. Kernel density based estimation of the measures is also considered, and the use of the estimated functionals is illustrated in the analysis of two real data sets.